Existence of Weak Solutions for a Class of Quasilinear Parabolic Problems in Weighted Sobolev Space ()
1. Introduction
Now we consider the initial/boundary-value problem [1] as following
(1.1)
where
, 
, 
is a real positive parameter and 
is (uniformly) parabolic, 
for some fixed time
, 
is an open bounded subset with smooth boundary in
, 
is given, 
is the unknown, 
, 
, 
are functions satisfying some suitable conditions [2-4].
The main purpose of this paper is to establish the existence of weak solutions for the parabolic initial/boundary-value problem (1.1) in a weighted Sobolev space. For this purpose, we assume for now that
1) 
is a positive measurable sufficiently smooth function2) 
is a non-negative smooth function which may change sign3) 
is a weighted Sobolev space [5-8] with a weight function
, its norm defined as
.
For convenience, we will denote 
by Xnote 
by
, and unless otherwise statedintegrals are over
.
Similar problems have been studied by Evans [9], he investigated the solvability of the initial/bondary-value problem for the reaction-diffusion system
(1.2)
Here
, 
, and as usual
, 
is open and bounded with smooth boundary. Via the techniques of Banach's fixed point theorem method, he obtained the existence and uniqueness and some estimates of the weak solution under the assumer that the initial function 
belongs to 
and 
is Lipschitz continuous. He also studied the nonlinear heat equation with a simple quadratic nonlinearity
(1.3)
The Blow-up solution has been established under the assumer that 
and 
are large enough in an appropriate sense.
The main results of this paper can be stated as followsTheorem 1.1. There exists a unique weak solution of problem (1.1) on the interval 
for the fixed time
.
For the further argument, we need the following Lemma.
Lemma 1.1. If
, then1) 
are the compact imbedding [6]2) 
are also compact imbedding.
Proof. 1) Since
, and 
is a positive sufficiently smooth function, there exists a positive constant C, such that
. Hence
for all
, and a.e. time
. We used the poincare’s inequality in the last inequality above. Thus1) Holds and is compact.
2) The proof of 2) is almost the same as 1). This completes the proof of Lemma 1.1.
2. Weak Solutions
According to Lemma 1.1, it suffices to consider the initial/boundary-value problem (1.1) in spaces 
and
. We will employ the Galerkin’s method to prove our results.
Definition 2.1. We say a function
is a weak solution of the parabolic initial/boundary-value problem (1.1) provided
1)
, for each
, and a.e. time
, and 2)
.
Here 
denotes the time-dependent bilinear form
for each 
and a.e. time
.
is the nonlinearity term. the pairing 
denoting inner product in
, 
being the pairing of 
and
.
By the Definition 2.1, we see
, and thus the equality 2) makes sense.
We now switch our view point, by associating with u a mapping
defined by
More precisely, assume that the functions 
are smooth1) 
is an orthogonal basis of 
and 2) 
is an orthogonal basis of
, (0 ≤ t ≤ T, i = 1, 2, ∙∙∙, m) taken with the inner product
Sm is the finite dimensional subspace spanned by
. Fix a positive integer m, we will look for a function 
of the form
(2.1)
Here we hope to select the coefficients
, (0 ≤ t ≤ T, i = 1, 2, ∙∙∙, m) such that
(2.2)
That is
(2.3)
This amounts to our requiring that um solves the “projection” of problem (1.1) onto the finite dimensional subspace
.
Theorem 2.1. (construction of approximate solutions)
For each integer 
there exists a function um of the form (2.1) satisfying the identities (2.3).
Proof. Taking 
arbitrary, then
Thus, 
, and
Hence,
(2.4)
Since 
is random, therefore, system (2.4) becomes
(2.5)
This is a nonlinear system of ordinary differential equation, according to the existence theory for nonlinear ODE, there exists a unique local solution on interval 
for fixed time T > 0. That is, the initial/boundaryvalue problem (1.1) has a unique local weak solution on the interval
.
3. Energy Estimates
Theorem 3.1. There exists a constant C, depending only on 
and
, such that
(3.1)
for
Proof. We separate this proof into 3 steps.
Step 1. Multiply equality (2.2) by 
and sum for
, and then recall to (2.1) to find
(3.2)
Whereas,
and
for a.e. time
, here, 
, since 
is a smooth function.
Consequently (3.2) yields the inequality
(3.3)
Since
, that is
, then by Sobolev imbedding theorem, we obtain
, and moreover,
here k is the best Sobolev constant [10-13].
Thus, we can write inequality (3.3) as
(3.4)
For a.e. time
, and appropriate constant
.
In addition, since
, by Sobolev interpolation inequality, we find
here
, and we have used the Young’s inequality with 
in the last inequality. Thus
(3.5)
By Lemma 1.1 2) and Sobolev’s inequality, we have
, 
is the best Sobolev imbedding constant, insert the inequality above and (3.5) into inequality (3.4) yields
(3.6)
for a.e. time
, and appropriate constants 
and
.
Furthermore, we rewrite inequality (3.6) as
(3.7)
for a.e. time
, and appropriate constants 
and
.
By Gronwall’s inequality, (3.7) yields the estimate
(3.8)
for a.e. time
, and appropriate constant
.
Step 2. Returning once more to inequality (3.7), we integrate from 0 to T and employ the inequality (3.8) to obtain
(3.9)
for a.e. time
, and appropriate constant C.
Step 3. Fix any
, with
, and write
, where 
and
. Since functions 
are orthogonal in
,
. Utilizing (2.2) we deduce for a.e. time
, that
Then (2.1) implies
Consequently,
since
. Thus
(3.10)
for a.e. time
, and appropriate constant C.
Combing (3.8), (3.9) and (3.10) we complete the proof of Theorem 3.1.
4. Existence of Weak Solutions
Next we pass to limits as
, to build a weak solution of our initial/boundary-value problem (1.1).
Theorem 4.1. There exists a local weak solution of problem (1.1).
Proof. According to the energy estimates (3.1), we see that the sequence 
is bounded in
, and 
is bounded in
. Consequently there exists a subsequence 
and a function
with
, such that
1) 
weakly in
, and 
strongly in
.
2) 
weakly in
.
Now we fix an integer 
and choose a function 
having the form
(4.1)
here 
are given smooth functions. We choose
, multiply (2.2) by
, sum
, and then integrate with respect to t, we find
(4.2)
We set
, and recall 1), 2) to find upon passing to weak limits that
(4.3)
This equality then holds for all functions 
, as functions of the form (4.1) are dense in this space. Hence in particular
(4.4)
for each 
and a.e. time
.
In order to prove
, we first note from (4.3) that
(4.5)
for each 
with
. Similarly, from (4.2) we deduce
(4.6)
We set 
and once again employ 1), 2), we obtain
(4.7)
since 
in
. As 
is arbitrary, comparing (4.5) and (4.7), we conclude
. This completes the proof of theorem 4.1.
5. Uniqueness of Weak Solutions
In this part, we will prove Theorem 1.1.
Proof. Let 
and 
are two weak solutions for the initial/boundary-value problem, put
, and insert it into the origin equation, we discover
Taking
, we obtain the energy estimates inequality
Since
,
. So we have 
for a.e. time
. This completes the proof of Theorem 1.1.
6. Conclusion
In this paper, we established the existence and uniqueness of weak solutions for initial/boundary-value parabolic problems with nonlinear perturbation term in weighted Sobolev space. First, we investigated the compact imbedding in weighted Sobolev space, which can be imbedded compactly into 
and 
spaces. By exploiting Sobolev interpolation inequalities and extending Galerkin’s method to a new class of nonlinear problems, we proofed the energy estimates of the equations and furthermore obtained the unique weak solution of the problem.
NOTES
#Corresponding author.