Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with p ( x , t ) Laplacian operator
Gao et al. Boundary Value Problems
Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with p(x, t )Laplacian operator
Yanchao Gao 0
Ying Chu 0
Wenjie Gao 1
0 School of Science, Changchun University of Science and Technology , Changchun, 130022 , P.R. China
1 School of Mathematics, Jilin University , Changchun, 130012 , P.R. China
The aim of this paper is to deal with the existence and nonexistence of weak solutions to the initial and boundary value problem for ut = div(?up(x,t)2?u + b(x, t)?u) + f (u). By constructing suitable function spaces and applying the method of Galerkin's approximation as well as weak convergence techniques, the authors prove the existence of local solutions. Furthermore, we choose a suitable testfunction, make integral estimates, and apply Gronwall's inequality to prove the uniqueness of weak solutions. At the end of this paper, the authors construct a suitable energy functional, obtain a new energy inequality, and apply a convex method to prove the nonexistence of solutions. Especially, it is worth pointing out that the results are obtained with the assumption that pt(x, t) is only negative and integrable, which is weaker than most of the other papers required.
nonstandard growth condition; nonexistence of solutions; Galerkin's approximation

with logarithmic module of continuity:
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1 Introduction
Consider the following initial and boundary value problem:
u(x, t) = ?,
??? u(x, ?) = u?(x),
?? ut = div(?up(x,t)???u + b(x, t)?u) + f (u), (x, t) ?
??
(x, t) = ?
x ? ,
? (?, T ) := QT ,
? (?, T ) := T ,
tinuous function satisfying
?z = (x, t), ? = (y, s) ? QT , z ? ?  < ?,
p(z) ? p(? ) ? ? z ? ?  ,
li?m?s?u+p ?(? ) ln ?? = C < +?,
and the coefficient b(x, t) is a Carath?odory function.
Model (?.?) proposed by R??i?ka may describe some properties of electrorheological
fluids which change their mechanical properties dramatically when an external electric
field is applied [?, ?]. The variable exponent p in Model (?.?) is a function of the external
electric field E? which is subject to the quasistatic Maxwell equations
Curl(E) = ?,
where ?? is the dielectric constant in vacuum and the electric polarization P is linear in E,
i.e. P = ?E. Another important application is the image processing where the anisotropy
and nonlinearity of the diffusion operator are used to underline the borders of the
distorted image and to eliminate the noise [?, ?]. For more physical background, the
interested reader may refer to [????].
In the case when p(x, t) is a fixed constant, there have been many results about the
existence, uniqueness, nonexistence, extinction of the solutions [??, ??, ??]. For nonconstant
case, the authors of [???] and the authors of [??] studied the existence and uniqueness
of weak solutions of the initial and Dirichlet boundary value problem with variable
exponent of nonlinearity. Besides, the authors of [??] applied the differential and variational
techniques to prove the existence of solutions when the exponent p only depends on the
spatial variable. Motivated by the work above, we consider the existence and uniqueness
of solutions to Problem (?.?). However, since the coefficient b(x, t) is degenerate or
singular, it is natural to ask: which kind of conditions on the coefficient b(x, t) guarantees that
Problem (?.?) admits a local solution? In our paper, we construct suitable function spaces
and apply Galerkin?s method to prove the existence of weak solutions to Problem (?.?) with
necessary uniform estimates and compactness argument. In addition, there exist some
difficulties such as the failure of the monotonicity of the energy functional, the anisotropy
of the diffusive operator and the gap between the norm and the modular, which make the
methods used in [??] fail. In order to overcome such difficulties, we have to search a new
technique or method. In this paper, by constructing a revised energy functional and
combining a new energy estimate with convex method, we obtain the nonexistence of weak
solutions when the exponents p is a function with respect to time and spatial variables.
Especially, it is worth pointing out that the results are obtained with the assumption that
pt(x, t) is only negative and integrable which is weaker than those the most of the other
papers required.
The outline of this paper is the following: In Section ?, we shall introduce the function
spaces of OrliczSobolev type, give the definition of the weak solution to the problem, and
prove the existence of weak solutions with Galerkin?s method and the uniqueness of the
solution. In Section ?, we establish sufficient condition of nonexistence of weak solutions
2 Existence of local solutions
u Vt( ) = u ?, + ?u p(?), ;
u H(QT ) = u ?,QT + ?u p(?),QT ,
up(?) dx < ? ,
u p(?) = inf ? > ?, Ap(?)(u/?) ? ? ;
u W?,p(?)( ) = u p(?), + ?u p(?), ;
and denote by H (QT ) the dual of H(QT ) with respect to the inner product in L?(QT ).
From [??], we know that condition (?.?) implies that M = {u : u ? W ?,p(x)( ), u = ? on ? }
is equivalent to W??,p(x)( ) (the closure of C??( ) in W ?,p(x)( )).
u p(?) < ? (= ?; > ?)
u p(?) ? ?
? Ap(?)(u) < ? (= ?; > ?);
+ ?
u pp(?) ? Ap(?)(u) ? u pp(?);
? +
u pp(?) ? Ap(?)(u) ? u pp(?);
u p(?) ? ?
Ap(?)(u) ? ?;
Ap(?)(u) ? ?.
uv dx ?
u p(?) v q(?) ? ? u p(?) v q(?).
Because of the degeneracy, Problem (?.?) does not admit classical solutions in general.
So we introduce weak solutions in the following sense.
Definition ?.? A function u(x, t) ? H(QT ) ? L?(?, T ; L?( )), b(x, t)?u ? L?(?, T ; L?( ))
is called a weak solution of Problem (?.?) if for every testfunction
?up(x,t)???u?? dx dt < ?,
t?
t?
On the other hand,
b(x, t)?u ? L? ?, T ; L?( ) ,
which imply that the definition of weak solutions is well defined.
The main theorem in this section is the following.
Theorem ?.? Suppose that the continuous function f (s) and the exponents p(x, t), ? satisfy
conditions (?.?)(?.?). If the following conditions hold:
Corollary ?.? Let the conditions of Theorem ?.? be fulfilled, then the solution u ? H(QT )
to Problem (?.?) satisfies the identity
= ?,
R
(H?) the function f (s) is decreasing in s ? .
Furthermore, we have the following comparison theorem.
Corollary ?.? (Comparison principle) Let u, v ? H(QT ) ? L?(?, T ; H??( )) be two bounded
weak solutions of Problem (?.?) such that u(x, ?) ? v(x, ?) a.e. in . If the nonlinearity
exponents and the function f (s) satisfy the conditions of Theorem ?.?, then u(x, t) ? v(x, t) a.e.
in QT .
3 Nonexistence of global weak solutions
In this section, we concentrate on the study of nonexistence of weak solutions to Problem
(?.?). For convenience, we first state that the function f (s) and the coefficient b(x, t) satisfy
the following conditions:
b(x, t) ? ?,
f (u) ? C(R),
bt(x, t) ? ?,
?(x, t) ? QT ;
f (u)u ? p+G(u) ? ?,
Definition ?.? A function u(x, t) is called a global solution to Problem (?.?) if ?T > ? the
following property holds:
Otherwise, we say that Problem (?.?) does not admit global weak solutions.
Theorem ?.? Assume that u(x, t) ? H(QT ) ? L?(?, T ; L?( )), b(x, t)?u ? L?(?, T ; L?( ))
is the local solution to Problem (?.?). If (?.?) and (?.?) are fulfilled and u? ? W??,p(x)( ),
p+ > ?, such that
G(u?) dx >
?u?? dx +
To prove Theorem ?.?, we need the following lemma.
Lemma ?.? Assume that u ? H(QT ) is the solution to Problem (?.?), then u(x, t) satisfies
the following relation:
?u(x, t) p(x) dx +
?u?(x) p(x) dx +
G u(x, t) dx
G u?(x) dx.
Proof Following the lines of the proof of Lemma ?.? and Theorem ?.? in [?], we know that
ut ? L?(QT ). Noting that
= ?up(x)???u?ut,
and using the idea of the proof of Lemma ? in [?], we arrive at the relation
After integrating over (?, t), it is obvious that Lemma ?.? holds.
Proof of Theorem ?.? Let
t? =
K (t) =
?b(x, ? )?u? ? ?up(x) + f (u)u dx + ??.
By H?lder?s inequality, we have
u?? dx =
? K (t) ?
Therefore by Lemma ?.?, we obtain the following inequality:
Thus by Schwarz?s inequality and the definition of K (t), we have
bt?u?
K (t)K (t) ?
? K (t)
? ? ?u? dx +
? ? ?up(x) dx
uf (u) ? p G(u) dx ? ?? p+ ? ?
+
?G(u?) ? b(x, ?)?u?? ?
?u?
Noticing p+ > ?, K (t) > ?, we conclude from (?.?), (?.?), (?.?) that
K (t)K (t) ?
? ?, for t ? (?, T ),
which implies
? ?, for t ? (?, T ).
? T .
This completes the proof of Theorem ?.?.
On one hand, a simple analysis shows that
pt
= ?up(x,t)???u?ut + p? ?up(x,t) ln ?up(x,t) ? ? ,
?u(x, t) p(x,t) dx +
?u?(x) p(x,?) dx +
b(x?, t) ?u(x, t) ? dx ?
G u(x, t) dx
G u?(x) dx
?up(x,t)
{?up?e} p?(x, t)
ln ?up(x,t) ? ? pt(x, t) dx
dx ?
The second inequality above follows from
? ?e ? s ln s ? ?,
? ? s ? ?.
Lemma ?.? follows from (?.?) and (?.?).
Our main result is as follows.
G(u?) dx >
?u?? dx +
?u?p(x,?) dx
then there exists T ? ? (?, T ] such that
Proof We argue by contradiction. Define
It is easy to verify that
t? =
K (t) =
K (t) ? ? K (t) ?
Therefore by Lemma ?.? and (?.??), we obtain the following inequality:
? K (t)
? ? ?up(x,t) dx
?G(u?) ? b(x, ?)?u?? ?
?u?p(x,?) dx
weaker than that of [?, ?].
putation shows that
pt (x, y, z, t) =
pt (x, y, z, t) dx dy dz dt =
pt (x, y, z, t) ?/ L?.
Competing interests
The authors declare that they have no competing interests.
Authors? contributions
All authors contributed equally to this paper and they read and approved the final manuscript.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11271154) and by the 985 program of Jilin
University.
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