Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 158, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SOLUTION TO A NONLINEAR BLACK-SCHOLES EQUATION
MARIA CRISTINA MARIANI, EMMANUEL K. NCHEUGUIM, INDRANIL SENGUPTA
Abstract. Option pricing with transaction costs leads to a nonlinear BlackScholes type equation where the nonlinear term reflects the presence of transaction costs. Under suitable conditions, we prove the existence of weak solutions
in a bounded domain and we extend the results to the whole domain using a
diagonal process.
1. Introduction
In a complete financial market without transaction costs, the celebrated BlackScholes model (1973) [4] provides not only a rational option pricing formula, but
also a hedging portfolio that replicates the contingent claim. In the Black-Scholes
analysis, it is assumed that hedging takes place continuously, and therefore, in a
market with proportional transaction costs, it tends to be infinitely expensive. So
the requirement of replicating the value of the option continuously has to be relaxed.
The first model in that direction was presented by Leland (1985) [14]. He assumes
that the portfolio is rebalanced at a discrete time δt fixed and that the transaction
costs are proportional to the value of the underlying; that is the costs incurred at
each step is κ|ν|S, where ν is the number of shares of the underlying bought (ν > 0)
or sold (ν < 0) at price S and κ is a constant depending on individual investors.
Leland derived an option price formula that is the Black-Scholes formula with an
adjusted volatility
r
2 κ 1/2
√
σ̂ = σ 1 +
.
π σ δt
Hoggard, Whalley and Wilmott [10] derived a model for portfolios of options in the
presence of transaction costs in 1994. We will outline the steps that they followed.
Let C(S, t) be the value of the option and Π be the value of the hedge portfolio.
Assume that the value of the underlying follows the random walk
δS = µSδt + σSφδt1/2 ,
where φ is drawn from a normal distribution, µ is a measure of the average rate
of growth of the asset price also known as the drift, and σ is a measure of the
2000 Mathematics Subject Classification. 91G80, 35B45, 58J35, 35D30.
Key words and phrases. Option pricing; Black-Scholes equation; Sobolev space;
Schaefer’s fixed point theorem.
c
2011
Texas State University - San Marcos.
Submitted August 20, 2010. Published November 28, 2011.
1
2
M. C. MARIANI, E. K. NCHEUGUIM, I. SENGUPTA
EJDE-2011/158
fluctuation (risk) in the asset prices, it corresponds to the diffusion coefficient.
Then the change in the value of the portfolio over the time step δt is given by
1
∂2C
∂C
∂C
∂C
δΠ = σS
− ∆ φδt1/2 +
σ 2 S 2 2 φ2 + µS
+
− µ∆S δt − κS|ν|
∂S
2
∂S
∂S
∂t
Let us consider the delta hedging strategy; that is choose the number of assets held
short at time t to be ∆ = ∂C
∂S (S, t). Therefore the number of assets to be traded
after δt is given by
∂C
∂C
∂2C
(S + δS, t + δt) −
(S, t) '
σSφδt1/2 .
∂S
∂S
∂S 2
So the expected transaction cost over a time step is
r
∂2C 2
E[κS|ν|] =
κσS 2 2 δt1/2 ,
π
∂S
and the expected change in the value of the portfolio is
r
∂C
2 ∂ 2 C 1
∂2C
E(δΠ) =
δt.
+ σ 2 S 2 2 − κσS 2
∂t
2
∂S
πδt ∂S 2
If the holder of the option expects to make as much from his portfolio as from a
bank account at a riskless interest rate r (no arbitrage), then
∂C δt.
E(δΠ) = r C − S
∂S
Hence the Hoggard, Whalley and Wilmott model for option pricing with transaction
costs is given by
r
∂C
2 ∂ 2 C 1 2 2 ∂2C
∂C
2
= 0,
(1.1)
+ σ S 2 + rS
− rC − κσS
∂t
2
∂ S
∂S
πδt ∂S 2
ν=
for (S, t) ∈ (0, ∞) × (0, T ), and the final condition
C(S, T ) = max(S − E, 0),
S ∈ (0, ∞)
for European call options with strike price E.
Note that equation (1.1) contents the usual Black-Scholes terms with an additional nonlinear term modelling the presence of transaction costs. Setting
1
x = log(S/E), t = T − τ / σ 2 , C = EV (X, τ ),
2
equation (1.1) becomes
∂2V
∂2V
∂V
∂V ∂V
+
+ (k − 1)
− kV = κ∗ 2 −
,
2
∂τ
∂x
∂x
∂x
∂x
for (x, τ ) ∈ R × (0, T ∗ ), with the initial condition
−
(1.2)
V (x, 0) = max(ex − 1, 0), x ∈ R,
p
where k = r/(σ 2 /2), κ∗ = κ 8/(πσ 2 δt) and T ∗ = σ 2 T /2. Next set
V (x, τ ) = ex U (x, τ ).
Then (1.2) yields
−
∂2U
∂U
∂2U
∂U
∂U +
+ (k + 1)
= κ∗ 2 +
,
2
∂τ
∂x
∂x
∂x
∂x
(x, τ ) ∈ R × (0, T ∗ )
(1.3)
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BLACK-SCHOLES EQUATIONS
3
with the initial condition
U (x, 0) = max(1 − e−x , 0).
The previous discussion motivates us to consider the following problem that
includes cost structures that go beyond proportional transaction costs
−
∂2U
∂U
∂U ∂ 2 U ∂U
,
+
+α
= βF
,
2
∂t
∂x
∂x
∂x ∂x2
(x, t) ∈ R × (0, T )
(1.4)
and
U (x, 0) = U0 (x),
x ∈ R,
(1.5)
where α and β are nonnegative constants.
Publications related to the above problem can be found in [2, 6, 16]. It is also
worth noting that such problems can be solved using the techniques used in [18, 19].
The goal of this paper is to show that the theoretical problem (1.4)-(1.5) has a strong
solution where the derivatives are understood in the distribution sense. We use the
following assumptions:
(H1) F : R × R → R+ is a continuous function,
(H2) F (p, q) ≤ |p| + |q|,
∂
2
2
2
(R), ∂x
(H3) For U ∈ Hloc
F (U, ∂U
∂x ) ∈ L (0, T ; Lloc (R)). Let BR = {x ∈ R :
∂
k
F (wk , ∂w
|x| < R}. Then if wk → w in L2 (0, T ; H01 (BR )), then ∂x
∂x ) →
∂
∂w
2
2
∂x F (w, ∂x ) in L (0, T ; L (BR )).
1
(R),
(H4) U0 ∈ Hloc
(H5) β < 1.
This work is organized as follows: In section 2 we review some notions of functional analysis that will be used later, then in section 3 we solve a similar problem
in a ball and finally in section 4 we construct a solution in the whole domain using
a diagonal process.
2. Definitions and notion
Function Spaces. Let Ω ∈ Rp , p ∈ N, be an open subset.
Definition 2.1. Suppose u and v ∈ L1loc (Ω), and α is a multi-index constant. v is
said to be the αth -weak partial derivative of u, denoted Dα u = v, if
Z
Z
uDα φ dx = (−1)|α|
vφ dx
for any test function φ ∈
Ω
Cc∞ .
Ω
Note that a weak partial derivative of u, when it exists, is unique up to a set of
measure zero. The Sobolev space
H m (Ω) = {u ∈ L2 (Ω) : Dα u ∈ L2 (Ω), for any multiindex α with |α| ≤ m},
where the derivatives are taken in the weak sense, is a Hilbert space when endowed
with the inner product
X
(u, v)H m (Ω) =
(Dα u, Dα v)L2 (Ω) .
|α|≤m
Let
= {u ∈ H such that u = 0 on ∂Ω} be the closure of Cc∞ in H 1 (Ω). Let
the space H −1 (Ω) is the topological dual of H01 (Ω).
H01 (Ω)
1
4
M. C. MARIANI, E. K. NCHEUGUIM, I. SENGUPTA
EJDE-2011/158
Let X be a Banach space and let T be a nonnegative integer.
L2 (0, T ; X) consists of all measurable functions u : (0, T ) → X with
Z T
1/2
kukL2 (0,T ;X) :=
ku(t)k2X dt
< ∞.
The space
0
2
Note that L (0, T ; X) is a Banach space endowed with the norm kukL2 (0,T ;X) .
The space C([0, T ]; X) consists of all continuous functions u : [0, T ] → X with
kukC([0,T ];X) := max ku(t)kX < ∞.
0≤t≤T
Note that C([0, T ]; X) is a Banach space endowed with the norm kukC([0,T ];X) .
Schaefer’s fixed point theorem. Let X be a real Banach space.
Definition 2.2. A nonlinear mapping A : X → X is said to be compact if for each
∞
bounded sequence {uk }∞
k=1 , the sequence {A[uk ]}k=1 is precompact; that is, there
∞
∞
exists a subsequence {ukj }j=1 such that {A[ukj ]}j=1 converges in X.
Theorem 2.3 (Schaefer’s fixed point Theorem). Suppose A : X → X is a continuous and compact mapping. Assume further that the set
{u ∈ X such that u = λA[u] for some 0 ≤ λ ≤ 1}
is bounded. Then A has a fixed point.
We will use the Schaefer’s fixed point theorem in order to show the existence of
a solution in a ball.
3. Solutions in bounded domains
Let BR = {x ∈ R : |x| < R} be the open ball centered at the origin with radius
R. Assume that U0 is suitable cut into bounded functions defined on BR and such
that (H1)-(H5) are satisfied in BR × [0, T ]. Set w = ∂U
∂x and consider an analogous
problem in BR × [0, T ] with zero Dirichlet condition on the lateral boundary.
−
∂w ∂ 2 w
∂w
∂
∂w +α
(x, t) ∈ BR × (0, T )
+
= β F w,
2
∂t
∂x
∂x
∂x
∂x
w(x, 0) = w0 (x) x ∈ BR
w(x, t) = 0,
(x, t) ∈ ∂BR × [0, T ].
(3.1)
(3.2)
(3.3)
Definition 3.1. A function w is said to be a weak solution of (3.1)-(3.3) if w ∈
2
−1
L2 (0, T ; H01 (BR )), ∂w
(BR )) and
∂t ∈ L (0, T ; H
Z
Z ∂w ∂φ
∂w
∂w ∂φ
∂φ φ+
+ αw
dx = −β
F w,
dx
(3.4)
∂t
∂x ∂x
∂x
∂x ∂x
BR
BR
for all φ ∈ H01 (BR ).
Remark 3.2 ([5, Thm. 3, sec. 5.9.2]). If w belongs to L2 (0, T ; H01 (BR )) and ∂w
∂t
belongs to L2 (0, T ; H −1 (BR )), then:
(i) w ∈ C([0, T ]; L2 (BR ));
(ii) the mapping t → kw(t)k2L2 (BR ) is absolutely continuous with
Z
d
∂w
kw(t)k2L2 (BR ) = 2
wdt a.e. 0 ≤ t ≤ T
(3.5)
dt
BR ∂t
EJDE-2011/158
BLACK-SCHOLES EQUATIONS
5
Theorem 3.3 (A priori estimate). If w is a weak solution of (3.1)-(3.3), then there
exists a positive constant C independent of w such that
∂w
−1
max kw(t)kL2 (BR ) + kwkL2 (0,T ;H01 (BR )) + k
k 2
≤ Ckw0 kL2 (BR ) .
0≤t≤T
∂t L (0,T ;H (BR ))
Proof. Choosing w(t) ∈ H01 (BR ) as the test function in (3.4), we obtain
Z Z
∂w
∂w ∂w
∂w ∂w ∂w
w+
+ αw
dx = −β
dx.
F w,
∂t
∂x
∂x
∂x
∂x ∂x
BR
BR
Therefore, by (3.5),
1 d
∂w 2
1
kw(t)k2L2 (BR ) + k
kL2 (BR ) + α
2 dt
∂x
2
Z
BR
∂w2
dx = −β
∂x
Z
F w,
BR
∂w ∂w
dx
∂x ∂x
and from (3.3),
Z
1 d
∂w 2
F w, ∂w ∂w dx.
kw(t)k2L2 (BR ) + k
kL2 (BR ) ≤ β
2 dt
∂x
∂x ∂x
BR
Using (H2), we obtain
Z ∂w ∂w 2 ∂w 2
1 d
+
dx.
kw(t)k2L2 (BR ) + k
kL2 (BR ) ≤ β
|w|
2 dt
∂x
∂x
∂x
BR
By the Cauchy-Schwartz inequality with > 0,
1 d
∂w 2
kw(t)k2 2
+k
k 2
2 dt Z L (BR )
∂xZ L (BR )
Z
∂w 2
1
∂w 2
| dx + |
| dx +
|w|2 dx .
≤β
|
4 BR
BR ∂x
BR ∂x
Since β < 1, choosing 1, yields
d
kw(t)k2L2 (BR ) + C1 kwk2H 1 (BR ) ≤ C2 kwk2L2 (BR )
0
dt
for a.e. 0 ≤ t ≤ T , and appropriate positive constants C1 and C2 .
Next we write η(t) := kw(t)k2L2 (BR ) , then by (3.7),
(3.6)
(3.7)
η 0 (t) ≤ C2 η(t), for a.e.0 ≤ t ≤ T.
The differential form of Gronwall inequality implies
η(t) ≤ eC2 t η(0)
Since η(0) =
kw(0)k2L2 (BR )
=
a.e.0 ≤ t ≤ T.
kw0 k2L2 (BR ) ,
kw(t)k2L2 (BR ) ≤ eC2 t kw0 k2L2 (BR ) .
Hence
max kw(t)kL2 (BR ) ≤ C11 kw0 k2L2 (BR ) ,
(3.8)
0≤t≤T
where C11 is some constant. To obtain a bound for the second term, we consider
(3.7), and integrate from 0 to T , obtaining
Z T
Z T
2
2
2
kw(T )kL2 (BR ) − kw0 kL2 (BR ) + C1
kwkH 1 (BR ) dt ≤ C2
kwk2L2 (BR ) dt.
0
0
0
Therefore,
kw(T )k2L2 (BR )
Z
+ C1
0
T
kwk2H 1 (BR ) dt
0
Z
≤ C2
0
T
kwk2L2 (BR ) dt + kw0 k2L2 (BR ) .
6
M. C. MARIANI, E. K. NCHEUGUIM, I. SENGUPTA
Hence
Z
T
C1
0
kwk2H 1 (BR ) dt ≤ C2
0
EJDE-2011/158
T
Z
0
kwk2L2 (BR ) dt + kw0 k2L2 (BR ) .
Using (3.8) we conclude that
kwkL2 (0,T ;H01 (BR )) ≤ C22 kw0 kL2 (BR ) ,
(3.9)
where C22 is a constant. Finally, to obtain a bound for the third term, by fixing
v ∈ H01 (BR ) with kvkH01 (BR ) ≤ 1. By (3.4), we have
Z
Z
Z ∂w
∂w ∂v
∂v ∂w ∂v
dx = −β
v dx +
+ αw
F w,
dx.
∂t
∂x
∂x
∂x
∂x ∂x
BR
BR
BR
Thus
Z
BR
∂w
v dx ≤ ∂t
Z
∂w ∂v
BR
∂x ∂x
+ αw
∂v dx + β ∂x
Z
F w,
BR
∂w ∂v
dx.
∂x ∂x
By Hölder inequality,
Z
Z
Z
∂v 2 1/2
∂w 2 1/2 ∂w
dx
dx
v dx ≤
BR ∂x
BR ∂x
BR ∂t
1/2
1/2 Z ∂v Z
2
2 dx
|w| dx
+α
BR ∂x
BR
Z
Z ∂v 2 1/2
∂w 2 1/2 dx
+
F w,
dx
.
∂x
BR
BR ∂x
Since kvkH01 (BR ) ≤ 1, using (H2) and the Poincaré inequality we deduce
Z
∂w
v dx ≤ C33 kw(t)kH01 (BR ) ,
BR ∂t
where C33 is some constant. So
∂w
(t)kH −1 (BR ) ≤ C33 kw(t)kH01 (BR ) .
k
∂t
Therefore,
Z T
Z T
∂w
k
kw(t)k2H 1 (BR ) dt = C33 kwk2L2 (0,T ;H 1 (BR )) .
(t)k2H −1 (BR ) dt ≤ C33
0
0
∂t
0
0
Then (3.9) implies
∂w
−1
k 2
≤ C33 kw0 k2L2 (BR ) .
(3.10)
∂t L (0,T ;H (BR ))
Take C = max{C11 , C22 , C33 } to obtain the required result in the Theorem.
k
Before proving the existence theorem in a ball, we state the following energy
estimate from the theory of linear parabolic partial differential equations.
Lemma 3.4 ([5, Theorem 2 page 354]). Consider the problem
∂w ∂ 2 w
∂w
−
−α
= f (x, t) in BR × (0, T )
2
∂t
∂x
∂x
w(x, 0) = w0 (x) on BR × {0}
w(x, t) = 0
on ∂BR × [0, T ]
with f ∈ L2 (0, T ; L2 (BR )) and w0 ∈ L2 (BR ).
(3.11)
EJDE-2011/158
BLACK-SCHOLES EQUATIONS
7
Then there exists a unique u ∈ L2 (0, T ; H01 (BR )) ∩ C([0, T ]; L2 (BR )) solution of
(3.11) that satisfies
∂u
max ku(t)kL2 (BR ) + kukL2 (0,T ;H01 (BR )) + k kL2 (0,T ;H −1 (BR ))
0≤t≤T
∂t
(3.12)
≤ C kf kL2 (0,T ;L2 (BR )) + kw0 kL2 (BR ) ,
where C is a positive constant depending only on R and T .
We need another Lemma that follows directly from [5, Theorem 5, page 360].
Lemma 3.5 (Improved regularity). Consider the problem
∂w ∂ 2 w
∂w
−
−α
= f (x, t) in BR × (0, T )
∂t
∂x2
∂x
w(x, 0) = w0 (x) on BR × {0}
w(x, t) = 0
on ∂BR × [0, T ]
with f ∈ L2 (0, T ; L2 (BR )) and w0 ∈ H01 (BR ). Then this problem has a unique weak
2
−1
(BR )).
solution u ∈ L2 (0, T ; H01 (BR )) ∩ C([0, T ]; L2 (BR )), with ∂u
∂t ∈ L (0, T ; H
Moreover,
∂u
u ∈ L2 (0, T ; H 2 (BR )) ∩ L∞ (0, T ; H01 (BR )),
∈ L2 (0, T ; L2 (BR )).
∂t
We also have the estimate
∂u
ess sup0≤t≤T ku(t)kH01 (BR ) + kukL2 (0,T ;H 2 (BR )) + k kL2 (0,T ;L2 (BR ))
∂t
(3.13)
≤ C 00 kf kL2 (0,T ;L2 (BR )) + kw0 kH01 (BR ) ,
where C 00 is a positive constant depending only on BR , T and the coefficients of the
operator L.
Next we show the existence of a solution in a ball by using the Schaefer’s fixed
point Theorem.
Theorem 3.6. If (H1)–(H5) are satisfied, then (3.1)-(3.3) has a weak solution
w ∈ L2 (0, T ; H01 (BR )) ∩ C([0, T ]; L2 (BR )).
∂
Proof. Given w ∈ L2 (0, T ; H01 (BR )), set fw (x, t) := β ∂x
F w, ∂w
∂x . By (H3), fw ∈
L2 (0, T ; L2 (BR )). From Lemma 3.4 there exists a unique v ∈ L2 (0, T ; H01 (BR )) ∩
C([0, T ]; L2 (BR )) solution of
∂v
∂2v
∂v
− 2 −α
= f (x, t) in BR × (0, T )
∂t
∂x
∂x
v(x, 0) = v0 (x) on BR × {0}
v(x, t) = 0
(3.14)
on ∂BR × [0, T ]
Define the mapping
A : L2 (0, T ; H01 (BR )) → L2 (0, T ; H01 (BR ))
by w 7→ A(w) = v, where v is derived from w via (3.14).
We now show that the mapping A is continuous and compact. We first prove
the continuity. Let {wk }k ⊂ L2 (0, T ; H01 (BR )) be a sequence such that
wk → w
in L2 (0, T ; H01 (BR )).
(3.15)
8
M. C. MARIANI, E. K. NCHEUGUIM, I. SENGUPTA
EJDE-2011/158
By the improved regularity (3.13), there exists a constant C 00 , independent of {wk }k
such that
(3.16)
kvk kL2 (0,T ;H 2 (BR )) ≤ C 00 kfwk kL2 (0,T ;L2 (BR )) + kw0 kH01 (BR ) ,
for vk = A[wk ], k = 1, 2, . . . . By (H3) as wk → w in L2 (0, T ; H01 (BR )), we must have
fwk (x, t) → fw (x, t) in L2 (0, T ; L2 (BR )). Therefore kfwk (x, t)kL2 (0,T ;L2 (BR )) →
kfw (x, t)kL2 (0,T ;L2 (BR )) . Then the sequence {kfwk kL2 (0,T ;L2 (BR )) }k is bounded and
sup kfwk kL2 (0,T ;L2 (BR )) ≤ C 000 ,
(3.17)
k
for a constant C 000 . Thus by (3.16) and (3.17) the sequence {vk }k is bounded
k
uniformly in L2 (0, T ; H 2 (BR )). Similarly it can be proved that { ∂v
∂t }k is uniformly
2
−1
bounded in L (0, T ; H (BR )). Thus by Rellich’s Theorem (see [7]) there exists a
subsequence {vkj }j ∈ L2 (0, T ; H01 (BR )) and a function v ∈ L2 (0, T ; H01 (BR )) with
vkj → v
in L2 (0, T ; H01 (BR )), as j → ∞.
(3.18)
Therefore,
∂v
Z
BR
kj
∂t
φ+
∂vkj ∂φ
∂φ + αvkj
dx =
∂x ∂x
∂x
Z
BR
fwkj (x, t)φ dx
for each φ ∈ H01 (BR ). Using (3.15) and (3.18) we see that
Z Z
∂v
∂v ∂φ
∂φ φ+
+ αv
dx =
fw (x, t)φ dx.
∂t
∂x ∂x
∂x
BR
BR
Thus v = A[w]. Therefore
A[wk ] → A[w] in L2 (0, T ; H01 (BR )).
The compactness result follows from similar arguments.
To apply Schaefer’s fixed point Theorem in L2 (0, T ; H01 (BR )) we need to show
that the set {w ∈ L2 (0, T ; H01 (BR )) : w = λA[w] for some 0 ≤ λ ≤ 1} is bounded.
This follows directly from the a priori estimate (Theorem 3.3) with λ = 1.
2
1
Remark 3.7. Theorem 3.6 shows that w = ∂u
∂x ∈ L (0, T ; H0 (BR )) solves problem
2
2
1
(3.1)-(3.3); so u ∈ L (0, T ; H0 (BR ) ∩ H (BR )) and is a strong solution of problem
(1.4)-(1.5) in the bounded domain BR × [0, T ] with zero Dirichlet condition on the
lateral boundary of the domain.
4. Construction of the solution in the whole domain
The next step is to construct a solution of (3.1)-(3.3) in the whole real line. To
do that, we approximate the real line by
R = ∪N ∈N BN = lim BN
N →∞
where BN = {x ∈ R : |x| < N }. We also approximate w0 by a sequence of
bounded function w0N defined in BN such that |w0N | ≤ |w0 | and w0N → w0 in
L2loc (R). For N ∈ N, there exists wN ∈ L2 (0, T ; H01 (BN )) ∩ C([0, T ]; L2 (BN )) with
∂wN
2
−1
(BN )), weak solution of
∂t ∈ L (0, T ; H
−
∂w
∂
∂w ∂w ∂ 2 w
+
+α
= β F w,
(x, t) ∈ BN × (0, T )
2
∂t
∂x
∂x
∂x
∂x
w(x, 0) = w0N (x) x ∈ BN
(4.1)
(4.2)
EJDE-2011/158
BLACK-SCHOLES EQUATIONS
w(x, t) = 0,
(x, t) ∈ ∂BN × [0, T ].
9
(4.3)
For any given ρ > 0, the following sequences are bounded uniformly for N > 2ρ:
{wN }N
in L2 (0, T ; H01 (Bρ )),
∂wN
}N in L2 (0, T ; H −1 (Bρ ))
∂t
Since these spaces are compactly embedded in L2 ((Bρ × (0, T )) then the sequence
{wN }N is relatively compact in L2 ((Bρ × (0, T )). By a standard diagonal process
we may select a subsequence also denoted {wN }N so that
{
wN → w
wN → w
a.e. in L2 (0, T ; L2loc (R)),
1
weakly in L2 (0, T ; Hloc
(R)).
Since F is continuous, passing to the limit in (3.4) yields that w is a weak solution
of (3.1)-(3.3) in R.
References
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10
M. C. MARIANI, E. K. NCHEUGUIM, I. SENGUPTA
EJDE-2011/158
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Maria Cristina Mariani
Department of Mathematical Sciences, The University of Texas at El Paso, Bell Hall
124, El Paso, TX 79968-0514, USA
E-mail address: mcmariani@utep.edu
Emmanuel Kengni Ncheuguim
Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM
88003-8001, USA
E-mail address: emmanou@nmsu.edu
Indranil SenGupta
Department of Mathematical Sciences, The University of Texas at El Paso, Bell Hall
316, El Paso, TX 79968-0514, USA
E-mail address: isengupta@utep.edu