Funkcialaj Ekvacioj, 39 (1996) 449-468
The Cauchy Problem for Degenerate Parabolic Equations and
Newton Polygon
By
Masahiro MIKAMI
(Ehime University, Japan)
1. Introduction
We are concerned with the homogeneous Cauchy problem for -evolution
equations:
$¥mathrm{p}$
(1)
$L(t, x, ¥partial_{t}, ¥partial_{X})u(t, ¥mathrm{x})¥equiv(¥partial_{t}^{m}-¥sum_{j=1}^{m}¥sum_{|a|¥leq pj}a_{ja}(t,x)¥partial_{X}^{a}¥partial_{t}^{m-j})u(t, x)=0$
(2)
$¥partial_{t}^{j-1}¥mathrm{u}(0,¥mathrm{x})=¥varphi_{j}(x)$
,
$1¥leq j¥leq m$
,
,
where $¥in[0, T]$ , $T>0$ and $p>0$ . It is known that the Cauchy problem is
wellposed if the equation (1) is -parabolic (see I. G. Petrowsky [1] and S.
Mizohata [2] . There are also several papers on the Cauchy problem for
, M.
degenerate parabolic equations published in the 1970’s (cf. O. A.
Miyake [4] and K. Igari [5] . The equations studied there are, however, of first
order in . Recently K. Kitagawa [6], [7] investigated necessary conditions
for the Cauchy problem $(1)-(2)$ to be well-posed. Put
$H^{¥infty}-$
$t$
$¥mathrm{p}$
$)$
$¥mathrm{O}¥mathrm{l}¥mathrm{e}¥dot{¥mathrm{i}}¥mathrm{n}¥mathrm{i}¥mathrm{k}$
$[3]$
$)$
$¥partial_{t}$
(3)
$a_{ja}(t, ¥mathrm{x})¥equiv t^{¥sigma(ja)}b_{ja}(t,x)$
plot the points
$(1+¥frac{¥sigma(ja)}{j},¥frac{|a|}{j})$
on the
with $P_{0}¥equiv(0,0)$ and
equation corresponding to each vertex
each side $P_{i}P_{i+1}(1¥leq i¥leq N-1)$ :
$P_{0}P_{1}¥cdots P_{N+1}$
,
-plane and draw the Newton polygon
$¥mathrm{X}¥mathrm{Y}$
$ P_{N+1}¥equiv$
$(¥infty,p)$
.
He defines a characteristic
and one corresponding to
$P_{i}(1¥leq i¥leq N)$
,
(4)
$p_{i}^{o}(x, ¥lambda, ¥xi)¥equiv¥lambda^{m}-¥sum_{(ja)¥in¥Gamma_{i}^{¥circ}}b_{ja}(0, x)(i¥xi)^{a}¥lambda^{m-j}=0$
(5)
$p_{i}(t,x, ¥lambda, ¥xi)¥equiv¥lambda^{m}-¥sum_{(ja)¥in¥Gamma_{i}}t^{¥sigma(ja)-q_{i}j}b_{ja}(0,¥mathrm{x})(i¥xi)^{a}¥lambda^{m-j}=0$
,
Masahiro MIKAMI
450
where
$P_{i}¥equiv(1+q_{i},r_{i})$
,
$¥Gamma_{i}^{o}¥equiv¥{(ja);(1+¥frac{¥sigma(ja)}{j},¥frac{|a|}{j})¥equiv P_{i}¥}$
and
$¥Gamma_{i}¥equiv¥{(ja);(1+¥frac{¥sigma(ja)}{j},¥frac{|a|}{j})¥in P_{i}P_{i+1}¥}$
.
Note that $p_{i}(0, x, ¥lambda, ¥xi)=p_{i}^{o}(x, ¥lambda, ¥xi)(1¥leq i¥leq N-1)$ . Regarding them as equaand by $¥lambda_{ik}(t,x, ¥xi)(1¥leq k¥leq m)$
tions of , denote their roots by
respectively.
$¥lambda_{ik}^{o}(x, ¥xi)$
$¥lambda$
Theorem 1 (K. Kitagawa [6, Theorem 1], [7, Theorem 1]).
wellposed, it is necessary that
Cauchy problem to be
For the
$H^{¥infty}-$
(6)
(7)
$¥mathrm{R}¥mathrm{e}¥lambda_{ik}^{o}(x, ¥xi)¥leq 0$
$Re¥lambda_{ik}(t, x, ¥xi)¥leq 0$
,
,
$(x, ¥xi)¥in R^{n}¥times R^{n}$
,
$1¥leq i¥leq N$
$(t, x, ¥xi)¥in(0, ¥infty)¥times R^{n}¥times R^{n}$
,
,
$1¥leq k¥leq m$
$1¥leq i¥leq N-1$
,
,
$1¥leq k¥leq m$
.
But, in his papers, nothing is mentioned about sufficient conditions. Note
first that the conditions (6) and (7) are not sufficient. In fact, consider the
equation:
,
(8)
$(¥partial_{t}-a¥partial_{X}^{2}-bt^{3}¥partial_{X}^{3}-ct^{5}¥partial_{¥mathrm{x}}^{4})u(t,x)=0$
$b¥neq 0$ .
In this example, (6) and
where , and are complex constants with
$c
¥
leq
0$
$a
¥
geq
0$
. But, by applying the theorem
and
(7) are equivalent to that
of Petrowsky, which will be cited in §3, we see easily that the Cauchy problem
$a>0$
-wellposed if and only if
for this equation with datum at $t=0$ is
$c<0$ .
and
In this paper, we prove
$a$
$b$
$¥mathrm{I}¥mathrm{m}$
$c$
$¥mathrm{R}¥mathrm{e}$
$¥mathrm{R}¥mathrm{e}$
$¥mathrm{R}¥mathrm{e}$
$¥mathrm{H}$
$¥mathrm{R}¥mathrm{e}$
Theorem 2. Suppose the following A.1-A.4:
depend only on
and
A.I the coefficients
$t¥in[0, T_{0}]$ , $T_{0}>0$ .
$a_{ja}$
$b_{ja}$
$t$
and are continuous in
Degenerate Parabolic Equations
A.2
A.3
are non-negative rational numbers.
$¥sigma(ja)$
there exist
(9)
451
$¥delta_{i}>0$
such that
$¥mathrm{R}¥mathrm{e}¥lambda_{ik}^{o}(¥xi)¥leq-¥delta_{i}|¥xi|^{r_{i}}$
,
$¥xi¥in R^{n}$
,
$1¥leq i¥leq N$
,
$1¥leq k¥leq m$
.
A.4
(10)
$Re¥lambda_{ik}(t, ¥xi)¥leq 0$
,
$(t, ¥xi)¥in(0, ¥infty)¥times R^{n}$
,
$1¥leq i¥leq N-1$
,
$1¥leq k¥leq m$
.
Then there exists $T>0$ such that the Cauchy problem $(1)-(2)$ is
wellposed,
$u(t,x)
¥
in
C_{t}^{m}([0,
T]$ ;
that is, for any
, there exists a unique solution
. Moreover $u(t, x)¥in C_{t}^{m}((0, T];H_{1/p}^{¥infty}(R^{n}))$, where
stands for the Gevrey class of exponent $1/p$ .
$H^{¥infty}-$
$¥varphi_{j}(x)¥in H^{¥infty}(R^{n})$
$H^{¥infty}(R^{n}))$
$H_{1/p}^{¥infty}(R^{n})$
The outline of the proof is as follows.
respect to , we get
$(¥subset H^{¥infty}(R^{n}))$
By the Fourier transform with
$x$
(11)
u(t, )
$(¥partial_{t}^{m}-¥sum_{j=1}^{m}¥sum_{|a|¥leq pj}a_{ja}(t)(i¥xi)^{a}¥partial_{t}^{m-j})$
$¥xi$
$=0$ .
The following proposition will be proved in §3.
Proposition 3. Suppose the conditions A.I-A.4.
constants , ,
and
such that any solution u
$T$
(12)
$C$
$¥gamma$
$(t, ¥xi)$
$¥rho$
Then there exist positive
of (11) satisfies
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C¥langle¥xi¥rangle^{¥gamma}¥exp¥{-¥rho t^{1+q_{N}}¥langle¥xi¥rangle^{p}¥}¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(0, ¥xi)|$
$(t, ¥xi)¥in[0, T]¥times R^{n}$
,
.
Theorem 2 immediately follows from this inequality.
To get the inequality (12), we need to obtain a series of energy estimates
corresponding to the vertexes and the sides of the Newton polygon. One of
the most crucial points is to prove the uniform boundedness of the definite
integral
(13)
$¥int_{a}^{b}|¥frac{d}{dt}¥lambda_{ik}(¥theta, ¥xi_{0})|d¥theta$
,
$1¥leq i¥leq N-1$
,
$1¥leq k¥leq m$
with respect to
(Lemma 5). We essentially use the fact that the
coefficients of the characteristic equations (5) are polynomials in
with some
$¥xi_{0}¥in S^{n-1}$
$t^{1/v}$
$v¥in N$
.
This paper is constituted as follows:
In §2, two algebraic lemmas are prepared, whose proofs are given in
§5. In §3, Proposition 3 is proved. In §4, we consider the uniform
$H^{¥infty}-$
452
Masahiro MIKAMI
wellposedness of the Cauchy problem and give sufficient conditions. We
should remark that the conditions A.1-A.4 are not sufficient for the uniform
-wellposedness.
Notation.
We use the following notation in this paper:
$X¥equiv(x_{1}, ¥ldots,x_{n})¥in R^{n}$
$¥sqrt{1+|¥xi|^{2}}$
,
$¥partial_{t}¥equiv¥frac{¥partial}{¥partial t}$
$|a|¥equiv a_{1}+¥cdots+a_{n}$
,
,
,
$¥xi¥equiv(¥xi_{1^{ }},¥ldots, ¥xi_{n})¥in R^{n}$
$¥partial_{x_{j}}¥equiv¥frac{¥partial}{¥partial x_{j}}$
$¥partial_{x}^{a}¥equiv¥partial_{x_{1}^{1}}^{a}¥ldots¥partial_{x_{n}^{n}}^{a}$
,
,
$|¥xi|¥equiv¥sqrt{¥xi_{1}^{2}++¥xi_{n}^{2}}¥cdots$
$a¥equiv(a_{1}, a_{2^{ }},¥cdots, a_{n})¥in N^{n}$
,
$ S^{n-1}¥equiv$
,
$¥{¥xi¥in R^{n}; |¥xi|=1¥}$
$m$
,
,
,
$H_{1/p}^{¥infty}(R^{n})¥equiv¥bigcup_{¥rho>0}¥{f(x)¥in H^{¥infty}(R^{n}); ¥exp(¥rho¥langle¥xi¥rangle^{p})¥hat{f}(¥xi)¥in L^{2}(R^{n})¥}$
denotes the set of
with value in $X$ .
$¥langle¥xi¥rangle¥equiv$
$N¥equiv¥{0,1,2, ¥ldots, ¥}$
$H^{¥infty}(R^{n})¥equiv¥bigcap_{s¥geq 0}¥{f(¥mathrm{x})¥in L^{2}(R^{n});¥langle¥xi¥rangle^{s}¥hat{f}(¥xi)¥in L^{2}(R^{n})¥}$
$C_{t}^{m}(I;X)$
,
,
times continuously differentiable functions of
$t¥in I$
2. Two algebraic lemmas
The following two lemmas are essential in our proof.
proved in §5.
Lemma 4.
Let
$A$
be an
$m¥times m$
matrix
of
They will be
form:
the following
$A=¥left(¥begin{array}{llll}a_{11} & 1 & & O¥¥a_{21} & & & ¥¥¥vdots & O & & 1¥¥a_{m1} & a_{m2} & ¥cdots & a_{mm}¥end{array}¥right)$
and
$¥lambda_{k}(1¥leq k¥leq m)$
$(n_{ij})_{1¥leq i,j¥leq m}$
(i)
(ii)
(iii)
its eigenvalues.
Then there exists a regular matrix $N=$
such that
$¥det N=1$ .
$n_{ij}$
are polynomials in
$(a_{ij},¥lambda_{k})$
$D¥equiv N^{-1}AN=¥left(¥begin{array}{llll}¥lambda_{1} & & & a_{ij}^{*}¥¥ & ¥lambda_{2} & & ¥¥O & & ..
Lemma 5.
Let
$P(¥lambda, t)$
& ¥lambda_{m}¥end{array}¥right)$
, where
be a polynomial in
$P(¥lambda, t)¥equiv¥lambda^{m}+¥sum_{j=1}^{m}a_{j}(t)¥lambda^{m-j}$
and denote continuous roots
.
of
,
are polynomials in
$a_{ij}^{*}$
$(¥lambda, t)$
by
.
, that is,
$a_{j}(t)¥equiv¥sum_{k=1}^{n}a_{jk}t^{k}$
$P(¥lambda, t)=0$
$(a_{ij}, ¥lambda_{k})$
and
$¥lambda_{k}(t)(1¥leq k¥leq m)$
.
$a_{jk}¥in C$
Let $M>0$ and
Degenerate Parabolic Equations
suppose
$|a_{jk}|¥leq M$
for
all
$j_{f}k$
.
Then
for any
453
$-¥infty<a<b<¥infty$ ,
there exists
$a$
positive constant $C=C(m, n, a, b,M)$ such that
(14)
$¥int_{a}^{b}|¥frac{d}{dt}¥lambda_{k}(t)|dt¥leq C(m, n, a, b, M)$
,
$1¥leq k¥leq m$
.
3. Energy estimate
In order to prove Proposition 3, we prepare two propositions. The
following proposition enables us to get energy estimates corresponding to the
vertexes of the Newton polygon. Let
and $¥sigma_{i}^{-}(1¥leq i¥leq N)$ stand for the
slopes of the sides
and those of the sides
respectively.
$¥sigma_{i}^{+}$
$P_{i}P_{i+1}$
$P_{i}¥_{}_{1} P_{i}$
Proposition 6. Assume (9), then there exist positive constants
such that estimates of the following form hold for any solution u(t, )
every
satisfying
, there exist positive
of (11): for
fixed constants
¥
constants
such that
$¥beta_{i}>0$
$(1¥leq i¥leq N)$
$¥xi$
$0<¥Theta_{i}¥leq¥beta_{i}$
$¥Theta_{i}$
$C^{o}=C^{o}( Theta_{i})$
(14)
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C^{o}(¥Theta_{i})|¥xi|^{¥mu_{i}^{o}}¥exp[-3K_{i}|¥xi|^{s_{i}}$
$¥times¥{(t|¥xi|^{¥sigma_{i}^{+}})^{1+q_{i}}-(¥Theta_{i}^{-1}|¥xi|^{-(¥sigma_{¥mathrm{i}}^{-}-¥sigma_{i}^{+})})^{1+q_{i}}¥}]$
$¥times¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(¥Theta_{i}^{-1}|¥xi|^{-¥sigma_{i}^{-}}, ¥xi)|$
,
,
$(t, ¥xi)¥in[¥Theta_{i}^{-1}|¥xi|^{-¥sigma_{i}^{-}}, ¥Theta_{i}|¥xi|^{-¥sigma_{¥mathrm{i}}^{+}}]¥times¥{¥xi¥in R^{n};|¥xi|^{-(¥sigma_{i}^{-}-¥sigma_{i}^{+})/2}2^{1/2(1+q_{i})}¥leq¥Theta_{i}¥}$
there
$¥mu_{i}^{o}¥equiv(m-1)(r_{i}-¥sigma_{i}^{-}q_{i})_{¥mathit{3}}$
$1¥leq i¥leq N$
$K_{i}¥equiv¥frac{¥delta_{i}}{4(1+q_{i})}$
and
$s_{i}¥equiv r_{i}-¥sigma_{i}^{+}(1+q_{i})$
.
$Proo/$.
To avoid confusion, omit a suffix .
$i$
,
$¥hat{u}_{j}^{0}(t, ¥xi)¥equiv t^{(m-j)q}|¥xi|^{(m-j)r}¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)$
$¥hat{U}^{o}(t, ¥xi)¥equiv¥prime t(¥hat{u}_{1}^{o_{ }},¥ldots,¥hat{u}_{m}^{O})$
,
$¥xi=¥xi_{0}|¥xi|$
Put
$1¥leq j¥leq m$
$(¥xi_{0}¥in S^{n-1})$
,
,
then
$¥partial_{t}¥hat{u}_{j}^{o}=t^{q}|¥xi|^{r}¥hat{u}_{j+1}^{o}+¥frac{(m-j)q}{t}¥hat{u}_{j}^{o}$
,
$1¥leq j¥leq m-1$
,
$¥partial_{t}¥hat{u}_{m}^{o}=¥sum_{j=1}^{m}[¥sum_{(ja)¥in¥Gamma^{o}}b_{ja}(0)(i¥xi_{0})^{a}+¥sum_{(ja)¥in¥Gamma^{o}}¥{b_{ja}(t)-b_{ja}(0)¥}(i¥xi_{0})^{a}$
$+¥sum_{(ja)¥not¥in¥Gamma^{o}}t^{¥sigma(ja)-q¥dot{|}}b_{ja}(t)(i¥xi_{0})^{a}|¥xi|^{|a|-rj]¥hat{u}_{m-j+1}^{o}t^{q}|¥xi|^{r}}$
.
,
454
Masahiro MIKAMI
Set
$¥partial_{t}¥hat{U}^{o}¥equiv[$
(
$.¥mathit{0}¥mathit{1}.$
$a_{1}^{o}O1)+¥left(¥begin{array}{lll} & O & ¥¥b_{m}^{o} & ¥cdots & b_{1}^{o}¥end{array}¥right)$
$0mo$
$..¥cdot..$
.
.
$+¥left(¥begin{array}{llll}d_{1}^{o} & & & O¥¥ & ¥ddots & & ¥¥O & & d_{m-1}^{o} & ¥¥c_{m}^{o} & ¥cdots & ¥cdots & c_{1}^{o}¥end{array}¥right)]t^{q}|¥xi|^{r}¥hat{U}^{o}$
$¥equiv[A^{o}(¥xi_{0})+B^{o}(t, ¥xi_{0})+C^{o}(t, ¥xi)]t^{q}|¥xi|^{r}¥hat{U}^{o}$
,
where
$a_{j}^{o}=a_{j}^{o}(¥xi_{0})¥equiv¥sum_{(ja)¥in¥Gamma^{o}}b_{ja}(0)(i¥xi_{0})^{a}$
,
$b_{j}^{o}=b_{j}^{o}(t, ¥xi_{0})¥equiv¥sum_{(ja)¥in¥Gamma^{o}}¥{b_{ja}(t)-b_{ja}(0)¥}(i¥xi_{0})^{a}$
,
$c_{j}^{o}=c_{j}^{o}(t, ¥xi)¥equiv¥sum_{(ja)¥not¥in¥Gamma^{o}}t^{¥sigma(ja)-qi}b_{ja}(t)(i¥xi_{0})^{a}|¥xi|^{|a|-rj}$
$d_{j}^{o}=d_{j}^{o}(t, ¥xi)¥equiv(m-j)qt^{-(1+q)}|¥xi|^{-r}$
Note that for the roots
$1¥leq j¥leq m$
.
of the characteristic equation
and they are eigenvalues of the matrix
.
$¥lambda_{k}^{o}(¥xi)(1¥leq k¥leq m)$
,
By the assumption A.3,
$p^{o}(¥lambda, ¥xi)=0$
,
,
$|¥xi|^{-r}¥lambda_{k}^{o}(¥xi)=¥lambda_{k}^{o}(¥xi_{0})$
$A^{o}(¥xi_{0})$
$¥mathrm{R}¥mathrm{e}¥lambda_{k}^{o}(¥xi_{0})¥leq-¥delta$
,
By Lemma 4, there exists a regular matrix
the following conditions:
.
(i) $¥det N^{o}¥equiv 1$ ,
.
are polynomials in
(ii)
$¥xi_{0}¥in S^{n-1}$
.
$N^{o}(¥xi_{0})¥equiv(n_{jk}^{o}(¥xi_{0}))_{1¥leq j,k¥leq m}$
satisfying
$¥xi_{0}¥in S^{n-1}$
$(a_{j}^{o}, ¥lambda_{k}^{o})$
$n_{jk}^{0}$
(iii)
, where
$D^{o}¥equiv N^{o-1}A^{o}N^{o}=¥left(¥begin{array}{llll}¥lambda_{1}^{o} & ¥lambda_{2}^{o} & & a_{jk}^{*}¥¥ & & ¥ddots & ¥¥O & & & ¥lambda_{m}^{o}¥end{array}¥right)$
in
Put
$(a_{j}^{o},¥lambda_{k}^{o})$
.
$a_{jk}^{*}$
are polynomials
Degenerate Parabolic Equations
455
$I_{¥epsilon}¥equiv¥left(¥begin{array}{ll}1¥ddots & O¥¥O & ¥epsilon^{m-1}¥end{array}¥right)$
and
$N_{¥epsilon}^{o}¥equiv N^{o}I_{¥epsilon}$
, then
$D_{¥epsilon}^{o}¥equiv N_{¥epsilon}^{o-1}A^{o}N_{¥epsilon}^{o}=¥left(¥begin{array}{llll}¥lambda_{1}^{o} & & & ¥epsilon^{k-j}a_{jk}^{*}¥¥ & ¥lambda_{2}^{o} & & ¥¥ & & ..
We shall determine the size of later.
, then
Put
$¥epsilon$
& ¥¥O & & & ¥lambda_{m}^{o}¥end{array}¥right)$
.
and
$W_{¥epsilon}^{o}¥equiv N_{¥epsilon}^{o-1}¥hat{U}^{o}¥equiv^{t}(w_{1^{ }}^{o},¥ldots, w_{m}^{o})$
$S_{¥epsilon}^{o}¥equiv|W_{¥epsilon}^{o}|^{2}¥equiv¥sum_{j=1}^{m}|w_{j}^{o}|^{2}$
$¥partial_{t}W_{¥epsilon}^{o}=(D_{¥epsilon}^{o}+N_{¥epsilon}^{o-1}B^{o}N_{¥epsilon}^{o}+N_{¥epsilon}^{o-1}C^{o}N_{¥epsilon}^{o})t^{q}|¥xi|^{r}W_{¥epsilon}^{o}$
.
Since
$¥mathrm{R}¥mathrm{e}$
$(D_{¥epsilon}^{o}W_{¥epsilon}^{o}, W_{¥epsilon}^{o})=¥mathrm{R}¥mathrm{e}$
$¥{¥sum_{j=1}^{m}(¥lambda_{j}^{o}w_{j}^{o}+¥sum_{k=j+1}^{m}¥epsilon^{k-j}a_{jk}^{*}w_{k}^{o},$ $w_{j}^{o})¥}$
$¥leq¥sum_{j=1}^{m}(¥mathrm{R}¥mathrm{e}¥lambda_{j}^{o})|w_{j}^{o}|^{2}+¥sum_{j=1}^{m}¥sum_{k=j+1}^{m}¥epsilon^{k-j}|a_{jk}^{*}||w_{k}^{o}||w_{j}^{o}|$
$¥leq$
where
$(1 -¥delta+ const.¥epsilon)S_{¥epsilon}^{o}$
,
is a positive constant independent of , we have
const,
$¥epsilon$
$¥frac{d}{dt}S_{¥epsilon}^{o}(t, ¥xi)=2¥mathrm{R}¥mathrm{e}¥{(D_{¥epsilon}^{o}W_{¥epsilon}^{o}, W_{¥epsilon}^{o})+(N_{¥epsilon}^{o-1}B^{o}N_{¥epsilon}^{o}W_{¥epsilon}^{o}, W_{¥epsilon}^{o})+(N_{¥epsilon}^{o-1}C^{o}N_{¥epsilon}^{o}W_{¥epsilon}^{o}, W_{¥epsilon}^{o})¥}t^{q}|¥xi|^{r}$
$¥leq$
{
$-2¥delta+$
Here estimate
First, since
$|B^{o}|$
,
const, $¥epsilon+2(|B^{o}|+|C^{o}|)|N_{¥epsilon}^{o-1}||N_{¥epsilon}^{o}|$
$|C^{o}|$
,
$|N_{¥epsilon}^{o}|$
and
in turn.
$|N_{¥epsilon}^{o-1}|$
$|b_{j}^{o}(t, ¥xi_{0})|¥leq¥max_{0¥leq t¥leq¥Theta}|b_{ja}(t)-b_{ja}(0)|¥sum_{(ja)}1$
for any
$¥epsilon>0$
$|b_{j}^{o}(t, ¥xi_{0})|¥leq¥frac{¥epsilon^{m}}{¥sqrt{m}}$
that is,
,
$t^{q}|¥xi|^{r}S_{¥epsilon}^{o}(t, ¥xi)$
Assume
and
, there exists a positive constant
}
$ t¥in$
.
$[¥Theta^{-1}|¥xi|^{-¥sigma^{-}}, ¥Theta|¥xi|^{-¥sigma^{+}}]$
$b_{ja}(t)¥in C^{0}([0, T_{0}];C)$
$¥beta_{1}=¥beta_{1}(¥epsilon)$
$0<¥Theta¥leq¥beta_{1}$
,
such that
,
.
456
Masahiro MIKAMI
$0¥leq¥Theta¥leq¥beta_{1}$
.
Second, since
$|c_{j}^{o}(t, ¥xi)|¥leq¥max_{0¥leq t¥leq¥Theta}|b_{ja}(t)|¥sum_{(ja)¥not¥in¥Gamma^{¥circ}}t^{¥sigma(ja)-q¥dot{f}}|¥xi|^{|a|-rj}$
and
$|a|-rj$
$-¥sigma^{¥pm}(¥sigma(ja)-qj)¥leq 0$
,
$(ja)¥not¥in¥Gamma^{o}$
,
we decompose and estimate the summation as follows:
const.
$|c_{j}^{o}(t, ¥xi)|¥leq$
$¥{¥sum_{(ja)¥not¥in¥Gamma^{o},¥sigma(ja)>qj}¥Theta^{¥sigma(ja)-qj}|¥xi|^{|a|-rj-¥sigma^{+}(¥sigma(ja)-qj)}+¥sum_{(ja)¥not¥in¥Gamma^{o},¥sigma(j¥alpha)=q¥dot{|}}|¥xi|^{|a|-rj}$
$+¥sum_{(ja)¥not¥in¥Gamma^{o},¥sigma(ja)<q¥dot{f}}¥Theta^{-(¥sigma(ja)-q¥dot{|})}|¥xi|^{|a|-rj-¥sigma^{-}(¥sigma(ja)-qj)¥}}$
$¥leq$
const.
Then for any
.
$¥{¥sum_{(ja)¥not¥in¥Gamma^{o},¥sigma(ja)¥neq qj}¥Theta^{|¥sigma(ja)-q¥dot{¥mathfrak{s}}|}+(ja)¥not¥in¥Gamma^{o},¥sigma(ja)=q¥sum_{¥dot{f}}|¥xi|^{|a|-rj}¥}$
$¥epsilon>0$
, there exists a positive constant
$|c_{j}^{o}(t, ¥xi)|¥leq¥frac{¥epsilon^{m}}{¥sqrt{2m-1}}$
,
$¥beta_{2}=¥beta_{2}(¥epsilon)$
such that
$|¥xi|^{-(¥sigma^{-}-¥sigma^{+})/2}2^{1/2(1+q)}¥leq¥Theta¥leq¥beta_{2}$
.
Next, since
$|d_{j}^{o}(t, ¥xi)|¥leq(m-1)q¥Theta^{1+q}|¥xi|^{¥sigma^{-}(1+q)-r}¥leq(m-1)q¥Theta^{1+q}$
for any
$¥epsilon>0$
, there exists a positive constant
$|d_{j}^{o}(t, ¥xi)|¥leq¥frac{¥epsilon^{m}}{¥sqrt{2m-1}}$
,
such that
$¥beta_{3}=¥beta_{3}(¥epsilon)$
$0<¥Theta¥leq¥beta_{3}$
.
The estimates above imply that
$|C^{o}(t, ¥xi)|=¥sqrt{¥sum_{j_{-}^{-}1}^{m}¥{|c_{j}^{o}(t,¥xi)|^{2}+|d_{j}^{o}(t,¥xi)|^{2}¥}}¥leq¥epsilon^{m}$
$|¥xi|^{-(¥sigma^{-}-¥sigma^{+})/2}2^{1/2(1+q)}¥leq¥Theta¥leq¥min(¥beta_{2},¥beta_{3})$
Next, since
$n_{jk}^{o}(¥xi_{0})$
are bounded,
$|N_{¥epsilon}^{o}(¥xi_{0})|¥leq|N^{o}(¥xi_{0})||I_{¥epsilon}|¥leq$
const.
,
.
,
457
Degenerate Parabolic Equations
Moreover, since cofactors
of the matrix
$¥tilde{n}_{jk}^{o}(¥xi_{0})$
$|N_{¥epsilon}^{o-1}(¥xi_{0})|¥leq|I_{¥epsilon}^{-1}||N^{o-1}$
Put
$¥leq¥beta$
$N^{o}(¥xi_{0})$
$(¥xi_{0})|¥leq$
are bounded,
$const.¥epsilon^{1-m}$
Then, for any positive constant
, the following inequalities hold:
$¥beta¥equiv¥min$
$(¥beta_{1}, ¥beta_{2}, ¥beta_{3})$
.
$|B^{o}|¥leq¥epsilon^{m}$
$|N_{¥epsilon}^{o-1}|¥leq$
,
$|C^{o}|¥leq¥epsilon^{m}$
$const.¥epsilon^{1-m}$
,
,
$|N_{¥epsilon}^{o}|¥leq$
.
$¥Theta$
satisfying
$ 0<¥Theta$
and
const.
$|¥xi|¥geq(¥Theta 2^{-1/2(1+q)})^{-2/(¥sigma^{-}-¥sigma^{+})}$
.
Then
$¥frac{d}{dt}S_{¥epsilon}^{o}(t, ¥xi)¥leq$
$(-2¥delta+ const.¥epsilon)t^{q}|¥xi|^{r}S_{¥epsilon}^{o}(t, ¥xi)$
Since const. is independent of , we can choose the size of
$¥epsilon$
$¥epsilon$
satisfying
,
$¥frac{d}{dt}S_{¥epsilon}^{o}(t, ¥xi)¥leq-¥frac{3¥delta}{2}t^{q}|¥xi|^{r}S_{¥epsilon}^{o}(t, ¥xi)$
so we obtain
$S_{¥epsilon}^{o}(t, ¥xi)¥leq¥exp[¥int_{¥Theta^{-1}|¥xi|^{-¥sigma^{-}}}^{t}(-¥frac{3¥delta}{2}x^{q}|¥xi|^{r})dx]S_{¥epsilon}^{o}(¥Theta^{-1}|¥xi|^{-¥sigma^{-}}, ¥xi)$
.
Then
,
$S_{¥epsilon}^{o}(t, ¥xi)¥leq¥exp[-6K|¥xi|^{s}¥{(t|¥xi|^{¥sigma^{+}})^{1+q}-(¥Theta^{-1}|¥xi|^{-(¥sigma^{-}-¥sigma^{+})})^{1+q}¥}]S_{¥epsilon}^{o}(¥Theta^{-1}|¥xi|^{-¥sigma^{-}}, ¥xi)$
.
$(t, ¥xi)¥in[¥Theta^{-1}|¥xi|^{-¥sigma^{-}}, ¥Theta|¥xi|^{-¥sigma^{+}}]¥times¥{¥xi¥in R^{n};|¥xi|¥geq(¥Theta 2^{-1/2(1+q)})^{-2/(¥sigma^{-}-¥sigma^{+})}¥}$
Using
the inequalities
and replacing by
$S_{¥epsilon}^{o1/2}=|W_{¥epsilon}^{o}|¥leq|N_{¥epsilon}^{o-1}||¥hat{U}^{o}|$
$|N_{¥epsilon}^{o}|S_{¥epsilon}^{o1/2}$
Remark.
u,
and
$|¥hat{U}^{o}|¥leq|N_{¥epsilon}^{o}||W_{¥epsilon}^{o}|=$
we obtain (15).
In (15), consider in the case
, we are
$t=¥Theta|¥xi|^{-¥sigma^{+}}$
Noting the inequality
$|¥xi|^{-(¥sigma^{-}-¥sigma^{+})/2}2^{1/2(1+q)}¥leq¥Theta$
$-3K|¥xi|^{s}¥{¥Theta^{1+q}-(¥Theta^{-1}|¥xi|^{-(¥sigma^{-}-¥sigma^{+})})^{1+q}¥}¥leq-¥frac{3K}{2}¥Theta^{1+q}|¥xi|^{s}$
,
thus we obtain
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(¥Theta|¥xi|^{-¥sigma^{+}}, ¥xi)|¥leq C^{o}(¥Theta)|¥xi|^{¥mu^{o}}¥exp(-¥frac{3K}{2}¥Theta^{1+q}|¥xi|^{s})¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(¥Theta^{-1}|¥xi|^{-¥sigma^{-}}, ¥xi)|$
$|¥xi|¥geq(¥Theta 2^{-1/2(1+q)})^{-2/(¥sigma^{-}-¥sigma^{+})}$
,
.
The following proposition enables us to get energy estimates corresponding to the sides of the Newton polygon.
458
Masahiro MIKAMI
$(1 ¥leq i¥leq N-1)$
Proposition 7. Assume (10), then for the constants
given by Proposition 6, there exist positive constants $C=C(¥Theta_{i})$ and $R=R(¥Theta_{i})$
such that for any solution
of (11),
$¥Theta_{i}$
u
(16)
$(¥mathrm{t}, ¥xi)$
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C(¥Theta_{i})|¥xi|^{¥mu_{i}}¥exp(¥frac{K_{i}}{2}¥Theta_{i}^{1+q_{i}}|¥xi|^{s_{i}})¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(¥Theta_{i}|¥xi|^{-¥sigma_{i}^{+}}, ¥xi)|$
$(t, ¥xi)¥in[¥Theta_{i}|¥xi|^{-¥sigma_{i}^{+}}, ¥Theta_{i}^{-1}|¥xi|^{-¥sigma_{i}^{+}}]¥times¥{¥xi¥in R^{n};|¥xi|¥geq R(¥Theta_{i})¥}$
there
$¥mu_{i}¥equiv(m-1)(r_{i}-¥sigma_{i}^{+}q_{i})$
,
We omit a suffix
Proposition 6, put
,
, then set
Proof.
$¥theta¥equiv t|¥xi|^{¥sigma^{+}}$
,
,
$1¥leq i¥leq N-1$ .
$i$
$¥theta¥in$
also here.
For
$¥hat{u}_{j}^{o}(t, ¥xi)(1¥leq j¥leq m)$
$[¥Theta, ¥Theta^{-1}],¥hat{u}_{j}(¥theta, ¥xi)¥equiv¥hat{u}_{j}^{o}(¥theta|¥xi|^{-¥sigma^{+}}, ¥xi)$
and
given by
$(¥theta, ¥xi)¥equiv$
$t(¥hat{u}_{1},¥hat{u}_{2}, ¥ldots,¥hat{u}_{m})$
$¥partial_{¥theta}¥hat{u}_{j}=¥{¥hat{u}_{j+1}+(m-j)q¥theta^{-(1+q)}|¥xi|^{-s}¥hat{u}_{j}¥}¥theta^{q}|¥xi|^{s}$
$¥equiv¥{¥hat{u}_{j+1}+d_{j}(t, ¥xi)¥hat{u}_{j}¥}¥theta^{q}|¥xi|^{s}$
,
$1¥leq j¥leq m-1$
,
$¥partial_{¥theta}¥hat{u}_{m}=¥sum_{j=1}^{m}[¥sum_{(ja)¥in¥Gamma}¥theta^{¥sigma(ja)-q¥dot{|}}b_{ja}(0)(i¥xi_{0})^{a}+¥sum_{(ja)¥in¥Gamma}¥theta^{¥sigma(ja)-q}¥dot{|}¥{b_{ja}(¥theta|¥xi|^{-¥sigma^{+}})-b_{ja}(0)¥}(i¥xi_{0})^{a}$
$+¥sum_{(ja)¥not¥in¥Gamma}¥theta^{¥sigma(ja)-q¥dot{|}}b_{ja}(¥theta|¥xi|^{-¥sigma^{+}})(i¥xi_{0})^{a}|¥xi|^{|a|-rj-¥sigma^{+}(¥sigma(ja)-qj)}]¥hat{u}_{m-j+1}¥theta^{q}|¥xi|^{s}$
$¥equiv¥sum_{j=1}^{m}¥{a_{j}(¥theta, ¥xi_{0})+b_{j}(¥theta, ¥xi)+c_{j}(¥theta, ¥xi)¥}¥hat{u}_{m-j+1}¥theta^{q}|¥xi|^{s}$
.
Set
$¥partial_{¥theta}¥hat{U}=[¥left(¥begin{array}{lll}0 & 1 & O¥¥ & O & 1¥¥a_{m} & ¥cdots & a_{1}¥end{array}¥right)$
$+¥left(¥begin{array}{llll}d_{1} & & & O¥¥ & ..
$+$
$¥left(¥begin{array}{lll} & O & ¥¥b_{m} & ¥cdots & b_{1}¥end{array}¥right)$
& & ¥¥O & & d_{m-1} & ¥¥c_{m} & ¥cdots & ¥cdots & c_{1}¥end{array}¥right)]¥theta^{q}|¥xi|^{s}¥hat{U}$
$¥equiv[A(¥theta, ¥xi_{0})+B(¥theta, ¥xi)+C(¥theta, ¥xi)]¥theta^{q}|¥xi|^{s}¥hat{U}$
Now, note that
of the matrix
and they are eigenvalues
of the characteristic equation
$|¥xi|^{-r}¥lambda_{k}(t, ¥xi)=¥lambda_{k}(¥theta, ¥xi_{0})(1¥leq k¥leq m)$
$A(¥theta, ¥xi_{0})$
for the roots
$¥lambda_{k}(t, ¥xi)$
.
459
Degenerate Parabolic Equations
$p(t, ¥lambda, ¥xi)=0$
.
By the assumption A.4,
,
$(¥theta, ¥xi_{0})¥in[¥Theta, ¥Theta^{-1}]¥times S^{n-1}$
$¥mathrm{R}¥mathrm{e}¥lambda_{k}(¥theta, ¥xi_{0})¥leq 0$
By virtue of Lemma 4, there exists a regular matrix
satisfying the following conditions:
(i)
(ii)
,
are polynomials in
$¥det N¥equiv 1$
$n_{jk}$
(iii)
$[¥Theta, ¥Theta^{-1}]¥times S^{n-1}$
$(a_{j}, ¥lambda_{k})$
.
,
mine the size of
$t(w_{1}, w_{2}, ¥ldots, w_{m})$
, where
$a_{jk}^{*}$
are polynomials in
.
same
the
$N(¥theta, ¥xi_{0})¥equiv(n_{jk}(¥theta, ¥xi_{0}))_{1¥leq j,k¥leq m}$
.
$D¥equiv N^{-1}AN=¥left(¥begin{array}{lll}¥lambda_{1} & & a_{jk}^{*}¥¥ & ¥lambda_{2} & ¥¥O & & ¥lambda_{m}¥end{array}¥right)$
$(a_{j}, ¥lambda_{k})$
In
$(¥theta, ¥xi_{0})¥in$
.
way
as
$S_{¥epsilon}¥equiv|W_{¥epsilon}|^{2}¥equiv¥sum_{j=1}^{m}|w_{j}|^{2}$
$¥epsilon$
later.
6,
Proposition
and
put
$N_{¥epsilon}¥equiv NI_{¥epsilon}$
$D_{¥epsilon}¥equiv N_{¥epsilon}^{-1}AN_{¥epsilon}$
,
≡
$W_{¥epsilon}¥equiv N_{¥epsilon}^{-1}$
We shall deter-
.
Then
$¥partial_{¥theta}W_{¥epsilon}=¥{(D_{¥epsilon}+N_{¥epsilon}^{-1}BN_{¥epsilon}+N_{¥epsilon}^{-1}CN_{¥epsilon})¥theta^{q}|¥xi|^{s}+(¥frac{d}{d¥theta}N_{¥epsilon}^{-1})N_{¥epsilon}¥}W_{¥epsilon}$
,
$¥frac{d}{d¥theta}S_{¥epsilon}(¥theta, ¥xi)=2¥mathrm{R}¥mathrm{e}$ $[¥{(D_{¥epsilon}W_{¥epsilon}, W_{¥epsilon})+(N_{¥epsilon}^{-1}BN_{¥epsilon}W_{¥epsilon}, W_{¥epsilon})+(N_{¥epsilon}^{-1}CN_{¥epsilon}W_{¥epsilon}, W_{¥epsilon})¥}¥theta^{q}|¥xi|^{s}$
$+((¥frac{d}{d¥theta}N_{¥epsilon}^{-1})N_{¥epsilon}W_{¥epsilon}$
,
$W_{¥epsilon}$
)
$]$
$¥leq[¥{const.¥epsilon+2(|B|+|C|)|N_{¥epsilon}^{-1}||N_{¥epsilon}|¥}¥theta^{q}|¥xi|^{s}+2|¥frac{d}{d¥theta}N_{¥epsilon}^{-1}||N_{¥epsilon}|]S_{¥epsilon}(¥theta, ¥xi)$
.
Here and hereafter const, denotes a positive constant depending on . In the
, there exists a positive constant
same way as Proposition 6, for any
$R=R(¥Theta, ¥epsilon)$ such that the following inequalities hold:
$¥Theta$
$¥epsilon>0$
$|B|¥leq¥epsilon^{m}$
,
$|C|¥leq¥epsilon^{m}$
,
$|N_{¥epsilon}|¥leq$
const,
and
$|N_{¥epsilon}^{-1}|¥leq$
$const.¥epsilon^{1-m}$
,
$|¥xi|¥geq R(¥Theta,¥epsilon)$
Here note that we can obtain the estimates above for any constant
and $|a|-rj$ $-¥sigma^{+}(¥sigma(ja)-qj)<0$
Proposition 6 because ¥
$ sigma^{+}>0$
$(ja)¥not¥in¥Gamma$
.
$(¥neq 0)$
$¥Theta$
given by
when
$(¥neq 0)$
Since
$|¥frac{d}{d¥theta}N_{¥epsilon}(¥theta, ¥xi_{0})^{-1}|¥leq¥epsilon^{1-m}|¥frac{d}{d¥theta}N(¥theta, ¥xi_{0})^{-1}|¥leq$
$¥frac{d}{d¥theta}S_{¥epsilon}(¥theta, ¥xi)¥leq¥{const.¥epsilon|¥xi|^{s}+$
.
,
$const.¥epsilon^{1-m}(1+¥sum_{j=1}^{m}|¥frac{d}{d¥theta}¥lambda_{j}(¥theta, ¥xi_{0})|)$
,
$const.¥epsilon^{1-m}(1+¥sum_{j=1}^{m}|¥frac{d}{d¥theta}¥lambda_{j}(¥theta, ¥xi_{0})|)¥}S_{¥epsilon}(¥theta, ¥xi)$
460
Masahiro MIKAMI
then
$S_{¥epsilon}(¥theta, ¥xi)¥leq¥exp[¥int_{¥Theta}^{¥Theta^{-1}}¥{const.¥epsilon|¥xi|^{s}+$ $const.¥epsilon^{1-m}(1¥dotplus¥sum_{j=1}^{m}|¥frac{d}{d¥theta}¥lambda_{j}(¥theta, ¥xi_{0})|)¥}d¥theta]S_{¥epsilon}(¥Theta, ¥xi)$
.
By Lemma 5,
$S_{¥epsilon}(¥theta, ¥xi)¥leq¥exp¥{const.(¥epsilon|¥xi|^{s}+¥epsilon^{1-m})¥}S_{¥epsilon}(¥Theta, ¥xi)$
Thus, for the constant , we can take a small constant
energy estimate holds:
$¥Theta$
$¥epsilon$
.
such that the following
$S_{¥epsilon}(¥theta, ¥xi)¥leq¥exp(K¥Theta^{1+q}|¥xi|^{s}+const.)S_{¥epsilon}(¥Theta, ¥xi)$
,
$(¥theta, ¥xi)¥in[¥Theta, ¥Theta^{-1}]¥times¥{¥xi¥in R^{n};|¥xi|¥geq R(¥Theta, ¥epsilon)¥}$
Replacing the estimate above by
u,
.
we obtain (16).
of
Proposition 3. For the constants $¥beta_{i}(1¥leq i¥leq N)$ given by
Proposition 6, put
. Then we can choose a common constant
$T$ for
the energy estimates corresponding to the vertexes $P_{i}(1¥leq i¥leq N)$ and
the sides $P_{i}P_{i+1}(1¥leq i¥leq N-1)$ of the Newton polygon. First, note that, by
simple computation, there exists a positive constant
such that for any
solution
of (11),
$Proo/$
$T¥equiv¥min_{1¥leq i¥leq N}¥beta_{i}$
u
(17)
$C_{0}$
$(t, ¥xi)$
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C_{0}¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(0, ¥xi)|$
,
$(t, ¥xi)¥in[0, T^{-1}|¥xi|^{-¥sigma_{0}^{+}}]¥times R^{n}$
.
Combining Proposition 6, Proposition 7 and (17), we get the following estimates:
there exist positive constants $C$, and $M$ such that for any solution
of
(11),
u
$¥gamma$
(18)
$(¥mathrm{t}, ¥xi)$
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C|¥xi|^{¥gamma}¥exp[-¥sum_{v=0}^{i-1}K_{v}T^{1+qv}|¥xi|^{s_{v}}-3K_{i}|¥xi|^{r_{i}}$
,
$¥times¥{t^{1+q_{i}}-(T^{-1}|¥xi|^{-¥sigma_{i}^{-}})^{1+q_{i}}¥}]¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(0, ¥xi)|$
,
$(t, ¥xi)¥in[T^{-1}|¥xi|^{-¥sigma_{i}^{-}}, T|¥xi|^{-¥sigma_{i}^{+}}]¥times¥{¥xi¥in R^{n};|¥xi|¥geq M¥}$
(19)
,
,
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C|¥xi|^{¥gamma}¥exp(-¥sum_{v=0}^{i}K_{v}T^{1+q_{v}}|¥xi|^{s_{v}})¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(0, ¥xi)|$
$(t, ¥xi)¥in[T|¥xi|^{-¥sigma_{¥mathrm{i}}^{+}}, T^{-1}|¥xi|^{-¥sigma_{i}^{+}}]¥times¥{¥xi¥in R^{n};|¥xi|¥geq M¥}$
where
$1¥leq i¥leq N$
$q_{0}=-1$
,
$s_{0}=0$
and
$K_{0}$
is some real constant.
,
$1¥leq i¥leq N-1$
,
461
Degenerate Parabolic Equations
If
$t¥in[T^{-1}|¥xi|^{-¥sigma_{i}^{-}}, T|¥xi|^{-¥sigma_{¥mathrm{i}}^{+}}](1¥leq i¥leq N)$
, then
$t^{1+q_{i}}|¥xi|^{r_{i}}=(t^{1+q_{N}}|¥xi|^{p})(t^{q_{¥mathrm{i}}-q_{N}}|¥xi|^{r_{i}-p})$
,
$t^{q_{i}-q_{N}}|¥xi|^{r_{i}-p}¥geq(T|¥xi|^{-¥sigma_{i}^{+}})^{q_{i}-q_{N}}|¥xi|^{r_{i}-p}¥geq T^{q_{i}-q_{N}}$
If
$t¥in[T|¥xi|^{-¥sigma_{i}^{+}}, T^{-1}|¥xi|^{-¥sigma_{i}^{+}}](1¥leq i¥leq N-1)$
.
, then
$T^{1+q_{i}}|¥xi|^{s_{i}}=(t^{1+q_{N}}|¥xi|^{p})(t^{-1-q_{N}}T^{1+q_{i}}|¥xi|^{s_{i}-p})$
,
$t^{-1-q_{N}}T^{1+q_{i}}|¥xi|^{s_{i}-p}¥geq(T^{-1}|¥xi|^{-¥sigma_{i}^{+}})^{-1-q_{N}}T^{1+q_{i}}|¥xi|^{s_{i}-p}¥geq T^{2+q_{i}+q_{N}}$
.
Hence, by the inequalities (17), (18) and (19), there exists a positive constant
such that
(20)
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C|¥xi|^{¥gamma}¥exp$
$¥{-¥rho t^{1+q_{N}}|¥xi|^{p}¥}¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(0, ¥xi)|$
$(t, ¥xi)¥in[0, T]¥times¥{¥xi¥in R^{n};|¥xi|¥geq M¥}$
$¥rho$
,
,
that is, we get the inequality (12).
Theorem 2 immediately follows from Proposition 3 and the following
well-known theorem due to I. G. Petrowsky [1].
Petrowsky’s theorem A. Assume A.I. Then the Cauchy problem $(1)-(2)$ is
wellposed if and only if there exist two positive constants $C$ and such that
any solution
of (11),
$H^{¥infty}-$
for
(21)
$l$
u
$(t, ¥xi)$
,
$(t, ¥xi)¥in[0, T]¥times R^{n}$
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C¥langle¥xi¥rangle^{l}¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(0, ¥xi)|$
4. Uniform
$H^{¥infty}$
-wellposedness
Consider the non-homogeneous Cauchy problem instead of
(22)
.
$L(t, x, ¥partial_{t}, ¥partial_{x})u(t, x)=f(t, x)$
(23)
$¥partial_{t}^{j-1}u(t_{0}, x)=¥varphi_{j}(x)$
,
$(t, x)¥in[t_{0}, T]¥times R^{n}$
,
$1¥leq j¥leq m$
$(1)-(2)$
:
,
.
Following I. G. Petrowsky [1], we introduce
-wellposedness).
Definition 8 (uniform
formly-wellposed, if for any
$¥mathrm{H}^{¥infty}$
$¥varphi_{j}(x)¥in H^{¥infty}(R^{n})$
,
$1¥leq j¥leq m$
there exists a unique solution
,
$t_{0}¥in[0,$
$T)$
,
We say that (22)?(23) is uni$f(t,x)¥in C_{t}^{0}([0, T];H^{¥infty}(R^{n}))$
$u(t, x)¥in C_{t}^{m}([t_{0}, T];H^{¥infty}(R^{n}))$
of (22)?(23).
,
462
Masahiro MIKAMI
Concerning the uniform
$¥mathrm{H}^{¥infty}$
-wellposedness, we know
Petrowsky’s theorem B. Assume A.I. Then the Cauchy problem (22)?(23)
wellposed if and only if there exist two positive constants $C$ and
is uniformly
such that for any solution
of (11),
$H^{¥infty}-$
u
$l$
(24)
$(t, ¥xi)$
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C¥langle¥xi¥rangle^{l}¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t_{0}, ¥xi)|$
where
$C$
and
are independent of
$l$
$t_{0}$
,
$(t, ¥xi)¥in[t_{0}, T]¥times R^{n}$
,
.
The assumption A.1-A.4 are not sufficient for the uniform
wellposedness. The following example is due to K. Kitagawa [8].
$H^{¥infty}-$
Consider the operator
Example.
.
$L(t, ¥partial_{t}, ¥partial_{x})¥equiv¥partial_{t}-¥partial_{¥chi}^{¥mathit{6}}(t^{2}¥partial_{X}^{4}+2t¥partial_{X}^{2}+1)-t^{2}¥partial_{X}^{8}$
Drawing the Newton polygon, we see easily that the conditions (9) and (10) are
-wellposed
satisfied. Then, for the operator, the Cauchy problem $(1)-(2)$ is
-wellposed. In fact,
for sufficient small $T>0$ , however, is not uniformly
following to K. Kitagawa [8], putting $t=1/¥xi(¥xi-1)$ and $t_{0}=1/¥xi(¥xi+1)$
$(¥xi>1)$ for the solution of the equation
$¥mathrm{H}^{¥infty}$
$(¥partial_{t}+¥xi^{6}(t^{2}¥xi^{4}-2t¥xi^{2}+1)-t^{2}¥xi^{8})¥hat{u}(t, ¥xi)=0$
with datum at
u
(
$¥frac{1}{¥xi(¥xi-1)}$
$t=t_{0}$
,
, we have
$¥xi)=¥exp[¥int_{1/¥xi(¥xi+1)}^{1/¥xi(¥xi-1)}¥{-¥xi^{6}(t¥xi^{2}-1)^{2}+t^{2}¥xi^{8}¥}dt]$
$=¥exp¥{¥frac{4¥xi^{5}}{3(¥xi^{2}-1)^{2}}¥}¥hat{u}$
(
$¥frac{1}{¥xi(¥xi+1)}$
,
$¥xi$
u
$(¥frac{1}{¥xi(¥xi+1)},$ $¥xi)$
),
implying that the Cauchy problem is not uniformly
$¥mathrm{H}^{¥infty}$
-wellposed.
463
Degenerate Parabolic Equations
Now, replace the assumption A.4 by the following one:
A.4’
(25)
$Re¥lambda_{ik}(t, ¥xi)¥leq-¥delta_{i}|¥xi|^{r_{i}}$
,
$(t, ¥xi)¥in(0, ¥infty)¥times R^{n}$
Then, it is easy to obtain the estimate (24).
(26)
$¥mathrm{R}¥mathrm{e}¥lambda_{ik}(¥theta, ¥xi_{0})¥leq-¥delta_{i}$
,
,
$1¥leq i¥leq N-1$
In fact, setting
$(¥theta, ¥xi_{0})¥in(0, ¥infty)¥times S^{n-1}$
,
,
$1¥leq k¥leq m$
$¥theta=t|¥xi|^{¥sigma_{i}^{+}}$
$1¥leq i¥leq N-1$
,
.
, we have
$1¥leq k¥leq m$
and, in the same way as the proof of Proposition 6, we can get the following
energy estimates instead of (16):
$¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(t, ¥xi)|¥leq C(¥Theta_{i})|¥xi|^{¥mu_{i}}¥exp[-3K_{i}|¥xi|^{s_{i}}¥{(t|¥xi|^{¥sigma_{i}^{+}})^{1+q_{i}}-¥Theta_{i}^{1+q_{i}}¥}]$
$¥times¥sum_{j=1}^{m}|¥partial_{t}^{j-1}¥hat{u}(¥Theta_{i}|¥xi|^{-¥sigma_{i}^{+}}, ¥xi)|$
,
$(t, ¥xi)¥in[¥Theta_{i}|¥xi|^{-¥sigma_{i}^{+}}, ¥Theta_{i}^{-1}|¥xi|^{-¥sigma_{i}^{+}}]¥times¥{¥xi¥in R^{n};|¥xi|¥geq R(¥Theta_{i})¥}$
,
$1¥leq i¥leq N-1$ .
Thus we obtain
Theorem 9. Suppose A.1-A.3 and A.4;. Then there exists $T>0$ such
wellposed.
that the Cauchy problem (22)?(23) is uniformly
$H^{¥infty}-$
5. Proof of lemmas
Proof of
Lemma 4.
and
that is,
$¥mathrm{x}¥equiv t(x_{1}, x_{2}, ¥ldots,x_{m})$
Put
$ A_{m}¥equiv A¥equiv$
(
$¥mathrm{a}_{1},¥mathrm{a}_{2}$
,
$¥ldots$
, a ). Let
$m$
an eigenvector corresponding to it.
be one of
Then
$¥lambda$
$¥lambda_{k}$
$A_{m}¥mathrm{x}=¥lambda ¥mathrm{x}$
,
$¥left¥{¥begin{array}{l}a_{11^{X}1}¥dagger X_{2}=¥lambda x_{1}¥¥a_{21}x_{1}+x_{3}=¥lambda x_{2}¥¥.¥cdot¥¥.¥cdot¥¥.¥cdot¥¥ a_{m-1,1^{X}1}+¥chi_{m}=¥lambda x_{m-1}¥¥a_{m1^{X}1}+a_{m2}¥mathrm{x}_{2}+a_{m3^{X}3}¥cdots+a_{mm^{X}m}=¥lambda x_{m}¥end{array}¥right.$
If we assume
$t(1,x_{2}, ¥ldots, ¥mathrm{x}_{m})$
, then
where
$x_{1}=0$
,
$t(0, ¥ldots, jl, ¥ldots,0)$
$¥mathrm{x}=0$
, thus we can choose an eigenvector
$x_{j+1}=¥lambda x_{j}-a_{j1}$
and
$(1 ¥leq j¥leq m-1)$ .
Now
$N_{m}¥equiv(¥mathrm{x},¥mathrm{x}+¥mathrm{e}_{2}, ¥ldots, ¥mathrm{x}+¥mathrm{e}_{m})¥equiv(¥mathrm{n}_{1}, ¥ldots,¥mathrm{n}_{¥mathrm{m}})$
put
, then
$¥mathrm{x}=$
$¥mathrm{e}_{¥mathrm{j}}¥equiv$
464
Masahiro MIKAMI
$¥det N_{m}=¥det(¥mathrm{x}, ¥mathrm{e}_{2}, ¥ldots,¥mathrm{e}_{¥mathrm{m}})=1$
,
$A_{m}N_{m}=(A_{m}x, A_{m}x+A_{m}e_{¥mathit{2}}, ¥ldots,A_{m}x+A_{m}e_{m})$
$=¥lambda(¥mathrm{X}_{ },¥ldots, ¥mathrm{X})+$
Each column vector
follows:
(
$¥mathrm{o},$
$¥mathrm{a}_{2}$
,
$¥ldots$
, a ).
$¥mathrm{m}$
is a linear combination of
$¥mathrm{a}_{¥mathrm{i}}(2¥leq j¥leq m)$
$¥mathrm{n}_{1}$
,
$¥ldots$
,
$¥mathrm{n}_{¥mathrm{m}}$
as
$¥mathrm{a}_{2}=¥mathrm{e}_{1}+a_{m2}¥mathrm{e}_{¥mathrm{m}}$
$=(¥mathrm{n}_{1}-¥sum_{j=2}^{m}x_{j}¥mathrm{e}_{j})+a_{m2}(¥mathrm{n}_{¥mathrm{m}}-¥mathrm{n}_{1})$
$=¥mathrm{n}_{1}-¥sum_{j=2}^{m}x_{j}(¥mathrm{n}_{¥mathrm{j}}-¥mathrm{n}_{1})+a_{m2}(¥mathrm{n}_{¥mathrm{m}}-¥mathrm{n}_{1})$
$=(¥sum_{j=1}^{m}x_{j}-a_{m2})¥mathrm{n}_{1}-¥sum_{j=2}^{m-1}x_{j}¥mathrm{n}_{¥mathrm{j}}+(-x_{m}+a_{m2})¥mathrm{n}_{¥mathrm{m}}$
,
$(x_{1}=1)$
$¥mathrm{a}_{¥mathrm{i}}=¥mathrm{e}_{¥mathrm{i}-1}+a_{mj}¥mathrm{e}_{¥mathrm{m}}$
$(3¥leq j¥leq m)$
$=-(1+a_{mj})¥mathrm{n}_{1}+¥mathrm{n}_{¥mathrm{i}^{¥_}1}+a_{mj}¥mathrm{n}_{¥mathrm{m}}$
.
Thus
$A_{m}N_{m}=N_{m}$
Then, set
$¥left(¥begin{array}{lllll}¥lambda & ¥lambda+¥sum_{j=1}^{m}x_{j}-a_{m2} & ¥lambda-1-a_{m3} & ¥cdots & ¥lambda-1-a_{mm}¥¥0 & -x_{2} & 1 & & O¥¥¥vdots & ¥vdots & & ..
$D_{m}¥equiv N_{m}^{-1}A_{m}N_{m}¥equiv(_{0}^{¥lambda}$
$A_{m-1}*$
),
where
& ¥¥0 & -x_{m-1} & O & & 1¥¥0 & -x_{m}+a_{m2} & a_{m3} & ¥cdots & a_{mm}¥end{array}¥right)$
$*$
.
denote polynomials in
.
has the same form with
and the $(m -1)¥times(m-1)$ matrix
$m=1$
$m$
it
is
true
;
assume
for
is
trivial
Now we use induction on . The claim
$ A_{m1}¥_$
$(a_{ij}, ¥lambda_{k})$
for $m-1$
exists
(
an
$¥cdot$
.
$(m ¥geq 2)$
.
Now put
$(m -1)¥times(m-1)$
$¥lambda_{m}*$
).
Put
$¥lambda=¥lambda_{1}$
.
Since
matrix
$T¥equiv¥left(¥begin{array}{ll}1 & 0¥¥0 & N_{m-1}¥end{array}¥right)$
$ N_{m1}¥_$
and
$A_{m}$
$N_{m}^{-1}A_{m}N_{m}=(_{0}^{¥lambda_{1}}$
such
that
$N_{m}^{*}¥equiv N_{m}T$
$A_{m-1}*)$
, there
$N_{m-1}^{-1}A_{m1}¥_ N_{m1}¥_=$
, then
$N_{m}^{*-1}A_{m}N_{m}^{*}=$
465
Degenerate Parabolic Equations
$¥left(¥begin{array}{ll}1 & 0¥¥0 & N_{m-1}^{-1}¥end{array}¥right)$
Thus
$N_{m}$
(
$01$
$A_{m-1}*$
)
$¥lambda_{2}$
$¥left(¥begin{array}{ll}1 & 0¥¥0 & N_{m-1}¥end{array}¥right)=(_{o}^{¥lambda_{1}}$
satisfies the conditions
$(¥mathrm{i})-(¥mathrm{i}¥mathrm{i}¥mathrm{i})$
.
$¥lambda_{m}*)$
.
.
be one of the continuous roots of the
Write $¥lambda(t)¥equiv x(t)+iy(t)$ and $P(x+iy, t)¥equiv P_{1}(x,y, t)+$
$iP_{2}(x,y, t)$ , where $P_{1}(x,y, t)$ and $P_{2}(x,y, t)$ are polynomials in $(x,y, t)$ with real
coefficients. Hereafter consider the polynomial ring $R[x,y, t]$ with coefficient
field $R$ .
First, we show that the resultant $Q(y, t)¥equiv R(P_{1}(¥cdot,y, t), P_{2}(¥cdot,y, t))¥not¥equiv 0$ ,
which is, by the Euclidean algorithm and Gauss’ lemma, equivalent to that
$P_{1}(x,y, t)$ and $P_{2}(x,y, t)$ have no common divisor as a polynomial including
. Consider in the case $m=$ even. We can also show in the case $m=odd$ in
the same way. Put
Proof of
equation
Lemma 5.
$P(¥lambda, t)=0$
Let
$¥lambda(t)$
.
$x$
$P_{1}(x,y, t)=P_{1}(x)¥equiv x^{m}+p_{1}x^{m-1}+p_{2}x^{m-2}+¥cdots+p_{m}$
$P_{2}(¥mathrm{x},y, t)=P_{2}(x)¥equiv q_{1}x^{m-1}+q_{2}x^{m-2}+¥cdots+q_{m}$
,
,
then
$order_{y}$ $p_{j}(y, t)¥{_{=j}^{¥leq j-1}$
$order_{y}$
$((j=¥mathrm{o}¥mathrm{d}¥mathrm{d})j=¥mathrm{e}¥mathrm{v}¥mathrm{e}¥mathrm{n})$
’
$q_{j}(y, t)¥left¥{¥begin{array}{l}=j(j=¥mathrm{o}¥mathrm{d}¥mathrm{d})¥¥¥leq j-1(j=¥mathrm{e}¥mathrm{v}¥mathrm{e}¥mathrm{n})¥end{array}¥right.$
and
.
$p_{1}q_{2}q_{1}1$
$Q(y, t)=|q_{1}1$
$.p_{2}p_{1}q_{2}.$
.
$q_{m-1}...p_{2}1....$
$|$
$p_{m-1}q_{m-1}..q_{m}p_{1}q_{1}..$
$p_{m-1}p_{m}.q_{m}p_{2}q_{2}.$
.
$..p_{m}...$
.
$p_{m-1}q_{m-1}$
.
$p_{m}q_{m}$
.
Denoting the highest order terms of
and those of
respectively, by simple computation, we have
$p_{j}(¥cdot, t)$
$q_{j}^{o}(¥cdot, t)$
$q_{j}(¥cdot, t)$
by
$p_{j}^{o}(¥cdot, t)$
and by
466
Masahiro MIKAMI
.
.
$.p_{2}^{o}00.$
.
$q_{1}^{o}001$
der
$¥mathrm{t}¥mathrm{t}¥mathrm{h}¥mathrm{e}¥mathrm{h}¥mathrm{i}¥mathrm{g}¥mathrm{h}¥mathrm{e}¥mathrm{s}¥mathrm{t}¥mathrm{o}¥mathrm{r}¥mathrm{e}¥mathrm{r}¥mathrm{m}¥mathrm{o}¥mathrm{f}Q(¥cdot,t)$
.
$=|q_{1}^{o}1$
$q_{m-1}^{¥circ}.....p_{2}^{o}1.......$
$.p_{m}^{o}p_{2}^{o}000$
$q_{m-1}^{o}...q_{1}^{o}000...$
$|$
.
$.....p_{m}^{o}........$
$q_{¥dot{m}-1}^{o}0...$
$p_{m}^{o}0$
.
.
¥
¥
¥
Now put $(x+iy)^{m}¥equiv P_{1}^{o}(x,y)+iP_{2}^{o}(x,y)$ and
, then
the determinant above is just equal to $Q^{o}(y)$ . Thus, “the highest order term of
$Q(¥cdot, t)’’=Q^{o}(¥cdot)$ .
The fact that
and
have some common divisor
as a polynomial including is equivalent to that $Q^{o}(y)¥equiv 0$ . By the theorem on
unique factorization in prime elements in polynomial ring, it is obvious that
and
have no common divisor including . Thus
that is, $Q(y, t)¥not¥equiv 0$ .
does not exist in
such that
Second, we take out values of
$Q^{o}(y) equiv R(P_{1}^{o}( cdot,y), P_{2}^{o}( cdot,y))$
$P_{1}^{o}(x,y)$
$P_{2}^{o}(x,y)$
$x$
$P_{1}^{o}(x,y)$
$P_{2}^{o}(x, y)$
$¥frac{d}{dt}y(t)$
$t$
$t¥in(a, b)$
By the theorem on implicit function, if
.
$Q^{o}(y)¥not¥equiv 0_{>}$
$x$
$¥frac{d}{dt}y(t)=-(¥frac{¥partial Q}{¥partial t}/¥frac{¥partial Q}{¥partial y})$
Then we take out values of
$t$
satisfying
,
$¥frac{¥partial}{¥partial y}Q(y, t)¥neq 0$
.
$¥frac{¥partial}{¥partial y}Q(y, t)=0$
is irreducible without loss of generality.
equations:
$Q(y, t)$
We can assume that
.
Consider the simultaneous
$¥left¥{¥begin{array}{l}Q(y,t)=0¥¥¥frac{¥partial}{¥partial y}Q(y,t)=0¥end{array}¥right.$
Now put
$T^{o}(t)¥equiv R(Q(¥cdot, t),¥frac{¥partial}{¥partial y}Q(¥cdot, t))$
.
$T^{o}(t)$
is a polynomial in
$t$
with real
coefficients, whose degree is depending only on $m$ and . If $T^{o}(t)¥equiv 0$, then
$Q(y, t)$ is reducible, thus
in $t¥in(a, b)$
. Then denote the zeros of
by , ,
, , where the integer has an upper bound depending only on $m$
and . Putting
$n$
$T^{o}(t)¥not¥equiv 0$
$T^{o}(t)$
$l^{o}$
$t_{1}^{o}$
$t_{2}^{o}$
$¥ldots$
$t_{l^{o}}^{o}$
$n$
$¥Delta^{o}$
:
$t_{0}^{o}¥equiv a<t_{l}^{o}<¥cdots<t_{l^{o}}^{o}<t_{l^{o}+l}^{o}¥equiv b$
,
467
Degenerate Parabolic Equations
then we have
$¥frac{d}{dt}y(t)=-(¥frac{¥partial Q}{¥partial t}/¥frac{¥partial Q}{¥partial y})¥in C^{0}((t_{k}^{o}, t_{k+1}^{o});R)$
Next, we take out values of
$t¥in(a, b)$
.
$t$
such that
,
$1¥leq k¥leq l^{o}$
$¥frac{d}{dt}y(t)$
.
changes the sign in
By the theorem on implicit function, if $¥frac{d}{dt}y(t)=0$, then
$¥frac{¥partial}{¥partial t}Q(y, t)=0$
.
Then consider the following simultaneous equations:
$¥left¥{¥begin{array}{l}Q(y,t)=0¥¥¥frac{¥partial}{¥partial t}Q(y,t)=0¥end{array}¥right.$
Putting
$T^{*}(t)¥equiv R(Q(¥cdot, t),¥frac{¥partial}{¥partial t}Q(¥cdot, t))$
in $t¥in(a, b)$ by , ,
bound depending only on $m$ and .
zeros of
$T^{*}(t)$
$t_{1}^{*}$
$t_{2}^{*}$
, in the same way as
$¥ldots$
$n$
$¥Delta^{*}:$
,
$t_{l^{*}}^{*}$
$T^{o}(t)$
, where the integer
, we denote the
$l^{*}$
has an upper
Put
$t_{0}^{*}¥equiv a<t_{1}^{*}<¥cdots<t_{l^{*}}^{*}<t_{l^{*}+1}^{*}¥equiv b$
and
$¥Delta¥equiv¥Delta^{o}¥cup¥Delta^{*}:$
then
$¥frac{d}{dt}y(t)$
$t_{0}¥equiv a<t_{1}<¥cdots<t_{l}<t_{l+1}¥equiv b$
is continuous and of definite sign in
$(l ¥leq l^{o}+l^{*})$
,
$t¥in(t_{k}, t_{k+1})(0¥leq k¥leq l)$
.
Thus
$¥int_{a}^{b}|¥frac{d}{dt}y(t)|dt=¥sum_{k=0}^{l}|¥int_{t_{k}}^{t_{k+1}}¥frac{d}{dt}y(t)dt|=¥sum_{k=0}^{l}|y(t_{k+1})-y(t_{k})|¥leq 2¥max_{t¥in[a,b]}|y(t)|¥cdot(l+1)$
Then there exists a positive constant
$C_{1}=C_{1}(m, n, a, b, M)$
$¥int_{a}^{b}|¥frac{d}{dt}y(t)|dt¥leq C_{1}(m, n, a, b, M)$
In the same way, there exists a positive constant
such that
.
$C_{2}=C_{2}(m, n, a, b, M)$
$¥int_{a}^{b}|¥frac{d}{dt}x(t)|dt¥leq C_{2}(m, n, a, b, M)$
.
such that
.
Hence the inequality (14) holds.
Acknowledgement. The author wishes to thank Professors K. Kitagawa
and K. Igari for their valuable suggestion and advice.
468
Masahiro MIKAMI
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$C^{¥infty}$
[7]
$C^{¥infty}$
[8]
nuna adreso:
Faculty of Engineering
Ehime University
Matsuyama 790
Japan
(Ricevita la 27-an de aprilo, 1995)