Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 96, pp. 1–6.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
HÖLDER CONTINUITY WITH EXPONENT (1 + α)/2 IN THE
TIME VARIABLE FOR SOLUTIONS OF PARABOLIC
EQUATIONS
JUNICHI ARAMAKI
Abstract. We consider the regularity of solutions for some parabolic equations. We show Hölder continuity with exponent (1 + α)/2, with respect to
the time variable, when the gradient in the space variable of the solution has
the Hölder continuity with exponent α.
1. Introduction
In this article we consider the Hölder continuity of solutions for the equation.
n
X
Lu :=
i,j=1
n
aij (x, t)
X
∂2u
∂u
∂u
=f
+
bi (x, t)
−
∂xi ∂xj
∂xi
∂t
i=1
in Q
(1.1)
where Q = Ω × (0, T ], Ω ⊂ Rn is a domain and T > 0. For the classical solution
u(x, t) of (1.1), we shall show the Hölder continuity with exponent (1 + α)/2 in the
time variable t, when the gradient of u with respect to the space variable x has
Hölder continuity with exponent α.
We assume that:
(H1) L is parabolic, i.e., for any (x, t) ∈ Q,
n
X
aij (x, t)ξi ξj > 0
for all 0 6= ξ = (ξ1 , . . . , ξn ) ∈ Rn .
i,j 1
Note that L is not necessary uniformly parabolic.
(H2) aij , bi ∈ C(Q) for i, j = 1, . . . , n where C(Q) denotes the space of continuous functions in Q.
(H3) There exist constants µ1 , µ2 > 0 such that
n
X
i=1
aii (x, t) ≤ µ1 ,
n
X
|bi (x, t)| ≤ µ2
for all (x, t) ∈ Q.
i=1
(H4) f = f (x, t) is a bounded continuous function in Q satisfying
|f (x, t)| ≤ µ3
for all (x, t) ∈ Q.
2010 Mathematics Subject Classification. 35A09, 35K10, 35D35.
Key words and phrases. Hölder continuity; parabolic equation.
c
2015
Texas State University - San Marcos.
Submitted February 16, 2015. Published April 13, 2015.
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J. ARAMAKI
EJDE-2015/96
In the following, for non-negative integers k, l and any set A ⊂ Rn , we denote the
space of functions u ∈ C(A × (0, T ]) such that u has continuous partial derivatives
∂xα u for |α| ≤ k and ∂tj u for j ≤ l in A × (0, T ] by C k,l (A × (0, T ]). Here
∂ |α| u
n
· · · ∂xα
n
Pn
for any multi-index α = (α1 , . . . , αn ) and |α| = i=1 αi . We also use the notation
ut = ∂t u, uxi = ∂xi u, uxi xj = ∂xi ∂xj u etc. Now we are in a position to state our
main result.
∂xα u =
1
∂xα
1
Theorem 1.1. Under the hypotheses (H1)–(H4), let u ∈ C 2,1 (Q) be a solution of
(1.1) in Q. Assume that there exist α ∈ (0, 1] and constants C1 , C2 ≥ 0 such that
|∇u(x, t) − ∇u(y, t)| ≤ C1 |x − y|α
(1.2)
for all (x, t), (y, t) ∈ Q, and
|∇u(x, t)| ≤ C2
(1.3)
for all (x, t) ∈ Q. Here and hereafter ∇ denotes the gradient operator with respect
to the space variable x.
(i) Let Ω0 ⊂ Ω be a subdomain such that dist(Ω0 , ∂Ω) ≥ d > 0, and define
0
Q = Ω0 × (0, T ]. Then there exist δ > 0 depending only on µ1 , µ2 , µ3 and α, K > 0
depending only on µ1 , µ2 , µ3 , d, α, C1 and C2 such that
|u(x, t) − u(x, t0 )| ≤ K|t − t0 |(1+α)/2
(1.4)
for all (x, t), (x, t0 ) ∈ Q0 with |t − t0 | < δ.
(ii) Furthermore, if we assume that ∂Ω 6= ∅ and u ∈ C 1,0 (Ω × (0, T ]) satisfies
that there exist β ∈ (0, 1] and a constant D ≥ 0 such that
|∇u(x, t) − ∇u(x, t0 )| ≤ D|t − t0 |(1+β)/2
for all x ∈ ∂Ω and t, t0 ∈ (0, T ], then for any σ > 0 there exists K > 0 depending
only on µ1 , µ2 , µ3 , C1 , C2 , D and σ such that
|u(x, t) − u(x, t0 )| ≤ K|t − t0 |(1+γ)/2 ,
γ = min{α, β}
for any (x, t), (x, t0 ) ∈ Q with |t − t0 | < σ.
Remark 1.2. Gilding [6] assumed that |u(x, t) − u(y, t)| ≤ C1 |x − y|α instead of
(1.2) and (1.3), and obtained
|u(x, t) − u(x, t0 )| ≤ K|t − t0 |α
instead of (1.4). Note that the papers of Brandt [4] and Knerr [7] can be viewed
as precursors to the present study. See also the discussion of Ladyzhenskaja et
al [8] in [7]. Then the author of [6] applied the result to the Cauchy problem for
the porous media equation in one dimension. See also Aronson [2] and Bénilan [3].
On the other hand, our result can be applied to the regularity for a quasilinear
parabolic type system associated with the Maxwell equation. For such application,
see Aramaki [1].
EJDE-2015/96
HÖLDER CONTINUITY
3
2. Proof of Theorem 1.1
We shall use a modification of the arguments in [6].
(i) Let Ω0 ⊂ Ω be a subdomain with dist(Ω0 , ∂Ω) ≥ d > 0 and define Q0 =
Ω0 × (0, T ]. Fix arbitrary points (x0 , t0 ), (x0 , t1 ) ∈ Q0 with 0 < t0 < t1 ≤ T and
choose 0 < ρ < d, and define µ and C so that
µ = max{µ1 , µ2 , µ2 C2 + µ3 }
and C =
C1
.
1+α
Moreover, we define a set and functions
N = {x ∈ Rn ; |x − x0 | < ρ} × (t0 , t1 ] ⊂ Q,
v ± (x, t) = µ{1 + 2sρ−2 (1 + ρ)}(t − t0 ) + sρ−2 |x − x0 |2 + Cρ1+α
± {u(x, t) − u(x0 , t0 ) − ∇u(x0 , t0 ) · (x − x0 )}
where “·” denotes the inner product in Rn . Let
s=
sup
t0 ≤t≤t1 ,x∈Ω0
|u(x, t) − u(x, t0 )|.
Since
vt± = µ{1 + 2sρ−2 (1 + ρ)} ± ut (x, t),
vx±i = 2sρ−2 (xi − x0,i ) ± {uxi (x, t) − uxi (x0 , t0 )},
vx±i xj = 2sρ−2 δij ± uxi xj (x, t)
where δij denotes the Kronecker delta, we have
Lv ± = −µ − 2sρ−2 µ(1 + ρ) + 2sρ−2
n
X
i=1
± Lu(x, t) ∓
n
X
aii (x, t) +
n
X
bi (x, t)(xi − x0,i )
i=1
bi (x, t)uxi (x0 , t0 )
i=1
≤ −µ − 2sρ−2 (µ + µρ) + 2sρ−2 (µ1 + µ2 ρ) + |f (x, t)|
+
n
X
|bi (x, t)||uxi (x0 , t0 )|
i=1
≤ −µ − 2sρ−2 (µ + µρ) + 2sρ−2 (µ1 + µ2 ρ) + µ3 + C2 µ2 ≤ 0.
Here we used the definition of µ.
(2.1)
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J. ARAMAKI
EJDE-2015/96
When t = t0 and |x − x0 | ≤ ρ, from the definition of C, we see that
v ± (x, t0 ) = sρ−2 |x − x0 |2 + Cρ1+α
± {u(x, t0 ) − u(x0 , t0 ) − ∇u(x0 , t0 ) · (x − x0 )}
= sρ−2 |x − x0 |2 + Cρ1+α
Z 1
(∇u(θx0 + (1 − θ)x) − ∇u(x0 , t0 )) · (x − x0 )dθ
±
(2.2)
0
−2
|x − x0 |2 + Cρ1+α
Z 1
− C1
|θx0 + (1 − θ)x − x0 |α dθ|x − x0 |
≥ sρ
0
C1 1+α
ρ
≥ 0.
1+α
When |x − x0 | = ρ and t0 < t ≤ t1 , using the definition of s, we can see that
−2
≥ sρ
|x − x0 |2 + Cρ1+α −
v ± (x, t) = µ{1 + 2sρ−2 (1 + ρ)}(t − t0 ) + s + Cρ1+α
± {u(x, t) − u(x0 , t0 ) − ∇u(x0 , t0 ) · (x − x0 )}
= µ{1 + 2sρ−2 (1 + ρ)}(t − t0 ) + s + Cρ1+α
± {u(x, t0 ) − u(x0 , t0 ) − ∇u(x0 , t0 ) · (x − x0 )}
(2.3)
± {u(x, t) − u(x, t0 )}
≥ µ{1 + 2sρ−2 (1 + ρ)}(t − t0 ) + s + Cρ1+α −
C1 1+α
ρ
−s
1+α
≥ 0.
Thus from (2.1), (2.2) and (2.3), we see that
Lv ± ≤ 0
±
v ≥0
in N,
(2.4)
on the parabolic boundary of N.
By the maximum principle (cf. Friedman [5, p. 34] or Lieberman [9, Chapter 2,
Lemma 2.3]), it follows that v ± ≥ 0 in N . Hence we have
∓ {u(x, t) − u(x0 , t0 ) − ∇u(x0 , t0 ) · (x − x0 )}
≤ Cρ1+α + µ{1 + 2sρ−2 (1 + ρ)}(t − t0 ) + sρ−2 |x − x0 |2 .
If we put x = x0 , then we see that
|u(x0 , t) − u(x0 , t0 )| ≤ Cρ1+α + µ{1 + 2sρ−2 (1 + ρ)}(t − t0 ).
Since x0 ∈ Ω0 and t ∈ (t0 , t1 ] are arbitrary, it follows that
s ≤ Cρ1+α + µ{1 + 2sρ−2 (1 + ρ)}(t1 − t0 )
(2.5)
1
= Cρ1+α + µ(t − t0 ) + s{4µρ−2 (1 + ρ)(t1 − t0 )}.
2
∗
Let ρ be the positive root of the quadratic equation y 2 = 4µ(1 + y)(t1 − t0 ), i.e.,
ρ∗ = 2µ(t1 − t0 ) + 2{µ(t1 − t0 ) + µ2 (t1 − t0 )2 }1/2 .
2
(2.6)
∗
If we define δ = d /(4µ(1 + d)), for t1 < t0 + δ, it is easily seen that ρ < d. Thus
we can replace ρ in (2.5) with ρ∗ . Therefore when t0 < t1 < t0 + δ, we see that
1+α
s ≤ C 2µ(t1 − t0 ) + 2{µ(t1 − t0 ) + µ2 (t1 − t0 )2 }1/2
EJDE-2015/96
HÖLDER CONTINUITY
5
1
+ µ(t1 − t0 ) + s
2
1+α
(t1 − t0 )(1+α)/2
= C 2µ(t1 − t0 )1/2 + 2{µ + µ2 (t1 − t0 )}1/2
1
+ µ(t1 − t0 )(1−α)/2 (t1 − t0 )(1+α)/2 + s.
2
Since t1 − t0 < δ, we have
1+α
+ µδ (1−α)/2 (t1 − t0 )(1+α)/2 .
s ≤ 2 C 2µδ 1/2 + 2{µ + µ2 δ}1/2
Thus we have
|u(x0 , t1 ) − u(x0 , t0 )| ≤ K(t1 − t0 )(1+α)/2
where
1+α
K = 2 C 2µδ 1/2 + 2{µ + µ2 δ}1/2
+ µδ (1−α)/2
for any t1 < t0 + δ. Since (x0 , t0 ) and (x0 , t1 ) with t0 < t1 ≤ T are arbitrary points
in Q0 , we get the conclusion of (i).
(ii) When (x0 , t0 ), (x0 , t1 ) ∈ Q with 0 < t0 < t1 < t0 + σ, we choose ρ∗ as in
(2.6). We define
N ∗ = {x ∈ Rn : |x − x0 | < ρ∗ } × (t0 , t1 ] ⊂ Rn × (0, T ],
w± (x, t) = v ± (x, t) + D(t1 − t0 )(1+β)/2 in N ∗ ∩ Q,
s=
sup
|u(x, t) − u(x, t0 )|.
t0 ≤t≤t1 ,x∈Ω
By a similar argument as in the proof of (i), we have
Lw± ≤ 0
w± ≥ 0
in N ∗ ∩ Q,
on the parabolic boundary of N ∗ ∩ Q.
If we choose µ = max{µ1 , µ2 , µ2 C2 + µ3 , Dσ (1+β)/2 }, from a similar argument as in
(i) we can get the conclusion of (ii).
Acknowledgments. We would like to thank the anonymous referee for his or her
very kind advice about a previous version of this article.
References
[1] J. Aramaki; Existence and uniqueness of a classical solution for a quasilinear parabolic type
equation associated with the Maxwell equation, to appear.
[2] D. G. Aronson, L. A. Caffarelli; Optimal regularity for one-dimensional porous medium flow,
Rev. Mat. Iberoamericana 2, (1986), 357-366.
[3] P. Bénilan; A strong Lp regularity for solutions of the porous media equation, in: Contributions to Nonlinear Partial Differential Equations (Eds. C. Bardos et al.), Reserch Notes in
Mathematics 89, Pitman, London, (1983), 39-58.
[4] A. Brandt; Interior Schauder estimates for parabolic differential- (or difference-) equations
via the maximum principle, Israel J. Math. 7, (1969), 254-262.
[5] A. Friedman; Partial Differential Equations of Parabolic Type, Princeton Hall Inc., Englewood
Cliffs, NJ, (1964).
[6] B. H. Gilding; Hölder continuity of solutions of parabolic equations, J. London Math. Soc.,
Vol. 13(2), (1976), 103-106.
[7] B. F. Knerr; Parabolic interior Schauder estimates by the maximum pronciple, Arch. Rational
Mech. Anal. 75, (1980), 51-58.
[8] O. A. Ladyzhenskaja, V. A. Solonnikov, N. N. Ural’ceva; Linear and Quasi-linear Equations
of Parabolic Type, Translations of Mathematical Monographs 23 AMS, Providence, RI, (1968).
[9] G. M. Lieberman; Second Order Parabolic Differential Equations, World Scientific, (2005).
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J. ARAMAKI
EJDE-2015/96
Junichi Aramaki
Division of Science, Faculty of Science and Engineering, Tokyo Denki University,
Hatoyama-machi, Saitama 350-0394, Japan
E-mail address: aramaki@mail.dendai.ac.jp