Existence of Weak Solutions for a Class of Quasilinear Parabolic Problems in Weighted Sobolev Space *

In this paper, we investigate the existence and uniqueness of weak solutions for a new class of initial/boundary-value parabolic problems with nonlinear perturbation term in weighted Sobolev space. By building up the compact imbedding in weighted Sobolev space and extending Galerkin’s method to a new class of nonlinear problems, we drive out that there exists at least one weak solution of the nonlinear equations in the interval   0,T 0 T    for the fixed time .


Introduction
Now we consider the initial/boundary-value problem [1] as following  is an open bounded subset with smooth boundary in N R  : is given, : T is the unknown, , , are functions satisfying some suitable conditions [2][3][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 function, 2) is a non-negative smooth function which may change sign, 3) is a weighted Sobolev space [5][6][7][8] with a weight function .
For convenience, we will denote 0 by X,  note Similar problems have been studied by Evans [9], he investigated the solvability of the initial/bondary-value problem for the reaction-diffusion system , , , m , g Here g g g   , 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   g x belongs to 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 follows, Theorem 1.1.There exists a unique weak solution of problem (1.1) on the interval   0,T 0 T  for the fixed time .
For the further argument, we need the following Lemma.
, and is a positive sufficiently smooth function, there exists a positive constant for all , and a.e.time .We used the poincare's inequality in the last inequality above.Thus, 1) Holds and is compact.
2) The proof of 2) is almost the same as 1).This completes the proof of Lemma 1.1.

Weak Solutions
According to Lemma 1.1, it suffices to consider the initial/boundary-value problem (1.1) in spaces   1 H 0  and .We will employ the Galerkin's method to prove our results.
is the nonlinearity term.the pairing denoting inner product in


being the pairing of and .
 1 , and thus the equality 2) makes sense.
We now switch our view point, by associating with u a mapping is an orthogonal basis of and Fix a positive integer m, we will look for a function


Here we hope to select the coefficients , .
This amounts to our requiring that u m solves the "projection" of problem (1.1) onto the finite dimensional subspace . .
there exists a function u m of the form (2.1) satisfying the identities (2.3).
Since is random, therefore, system (2.4) becomes 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 Proof.We separate this proof into 3 steps.
Step 1. Multiply equality (2.2) by m and sum for , and then recall to (2.1) to find for a.e.0 .
, then by Sobolev imbedding theorem, we obtain , and moreover, here k is the best Sobolev constant [10][11][12][13].Thus, we can write inequality (3.3) as , by Sobolev interpolation inequality, we find , and we have used the Young's inequality with in the last inequality.Thus 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 for a.e.time 0 t T   , and appropriate constant C.
) we e for a.e.time 0 t T   , that .  ;

Existence of Weak Solutions
and choose a fun weakly in Now we fix an integer N ction  , and ntegrate w unctions.We choose then i ith respect to t, we find In order to prove  

Conclusion
In per, we es blished the ex ique with lev space.First, we investigated the comweighted Sobolev space, which can into .this pa ta istence and un ness of weak solutions for initial/boundary-value parabolic problems nonlinear perturbation term in weighted Sobo pact imbedding in imbedded compac be tly   T for some fixed time T , 0  unless otherwise stated, integrals are over  .

 1 .
for fixed time T > 0. That is, the initial/boundaryvalue problem (1.1) has a unique local weak solution on the interval There exists a constant C, depending only on and and appropriate constant ing (3.8), (3.9) and (3.10) we complete the proof Next we pass to li its as m   , to build a weak blem

m solution of our initial/boundary-value pro 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  

 and once again employ 1 ), 2 u and 2 u
are two weak solutions for the initial/boundary-value problem, put u u  nd insert it into the origin equation, we disc Taking v u  , we obtain the energy es ates inequality

2 L
 spaces.Existence of P lass of QuasilinearBy 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.