Solutions with Dead Cores for a Parabolic P-Laplacian Equation

We study the solutions with dead cores and the decay estimates for a parabolic p-Laplacian equation with absorption by suband supersolution method. Special attention is given to the case where the solution of the steady-state problem vanishes in an interior region.


Introduction
In this paper, we study the following initial-boundary value problem for : (1.1)


The domain is smooth and bounded.
 > 1 N N    Our purpose is to describe how the solution   , u x t 1 < < 2 p of (1.1) tends to its steady state and the existence of dead cores.A dead core , i.e. a region where the solution vanishes identically may appear.Such a region is a waste from the engineering point of view.We concern its existence.Our method is the weak solution which is similar to that in [4] where the porous medium equation was considered.For the case of and of the problem (1.1), Chen, Qi and Wang [3] proved the existence of the singular solution.In [2], they also studied the long time behavior of solutions to the Cauchy problem of . For initial data of various decay rates, especially the critical decay , they showed that the solution converges as to a self-similar solution.

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We have known the following behavior of the absorption near : = 0 u < 1 st sorption, q .For small , the absorption is still relatively large in the case of strong absorption and will tend to drive the solution more quickly to zero than in the case of weak absorption.If is not identically zero in (1.1), the corresponding where From [1], we know that Problem (1.2) has a unique solution, a dead core exists if  is large enough and Proofs of existence and uniqueness for Problem (1.1) are based on a suitable notion of weak solution, which we include here for the sake of completeness.
Definition 1.1 Let Equation (1.5) is obtained easily by multiplying (1.1) by  , integrating over , and using the divergence theorem.
T Q A weak supsolution of (1.1) is defined by replacing the equal sign in (1.5) by and restricting   to be nonnegative.Similarly, we can define a weak subsolution.For our purposes, it suffices to consider the usual super-and subsolutions defined as follows.
We say that 0 u  is a supersolution of (1.1) if where in .
T It follows from the maximum principle that the solutions of (1.1) and of (1.2) satisfy in , in .
To show monotonicity of   , u x t in time, we need to impose a natural condition on the initial value This condition holds automatically if is a positive constant.If satisfies (1.7), it is an upper solution to (1.1) so that , is subsolution of (1.1), and hence

Monotonicity and Other Comparison Theorems
Consider problem (1.1) when only one part of the data is changed.We then have the following monotonicity properties: 1) Let 1 and 2 u be the solutions corresponding to 2) If the initial or the boundary value is decreased so is the solution.
and consider two domains These all are easy to prove using super-and subsolution techniques.Here we omit the proofs.
Next we look at the "lumped-parameter" problem and the steady-state problem with a view to using them as comparison problems for (1.1).The lumped parameter problem has no diffusion term.It can be obtained as a special case of (1.1) with initial value   0 1 u x  and boundary condition of vanishing normal derivative.We can then seek a solution independent of ( ) z t x : The solution is given explicitly by Therefore, extinction occurs in finite time if and only if .Then Comparison with (2.1) leads immediately to two results for (1.1).
, then is a supersolution of (1.1), so that and, if the absorption is strong, there is extinction in finite time for , then the solution   , z t  of (2.1) with initial value  is a subsolution of (1.1) so that    

  , > 0 u x t
The following theorem shows that (1.4) is necessary and sufficient for the existence of a dead core for sufficiently large  .
Proof.We shall construct, for  sufficiently large, an upper solution to (1.2) for a ball We begin by observing that on the positive real line, the function by where we use and 0 For any 0 , we can take 0 the distance from 0 x   = = R r x to the boundary.Theorem 2.2 shows that for  large enough, 0 x belongs to the dead core.This suggests making the following definition.
Definition 2.1 Let .Define The proof consists in noting that the function Thus, is a supersolution of (1.2) for any w , , 0 N p q p P r   w .Since vanishes at 0 x , so does  .This proves the first part of (2.4) and the second part follows at once.

The Corresponding Evolution Problem
Since (1.4) is satisfied with  large enough, the steady state has a dead core.Does the corresponding evolu-tion problem have a dead core and, if so, how does it behave for large t ?The answer is given by the fol-lowing theorem.Theorem 3.1 For fixed 0 , choose , > 0 for all t .Proof.Part (b) is equivalent to Theorem 2.1(b).To prove part (a), it suffices to exhibit a supersolution   , w x t such that   0 , = 0 w x t for .We try a function where   , z t  is the solution of the lumped--parameter problem with  to be choosen satisfies the differential inequality for a supersolution and  is the solution of the steady-state problem with 0 =   .Note that vanishes at 0 . From the definition of , we have w  , we obtain the desired result. .

Decay Estimates
We consider (1.1) with , where x  is the solution of the corresponding steady-state problem (1.2).It then follows, since  is a lower solution, that .If we assume in addition that , then is bounded a.e. in .
. If the data are smooth, the solution is continuous and the estimates hold pointwise.We seek decay estimates for . Our principal results are contained in the following theorem.
Theorem 4.1 is proved.
where     q t p q t q q t q q q q w div w w w z div w [4] C. Bandle, T.Nanbu, I. Stakgold, "Porous Medium Equation with Absorption," SIAM Journal on Mathematical Analysis, Vol. 29, 1998, pp. 1268-1278. doi:10.1137/S0036141096311423where we use .Because we also have and 1 q  w z  w   , we see that is a supersolution of (1.1).Then , where   t  satisfies (4.1) and  satisfies (1.2).Then, we have is a nondecreasing function with is an upper solution of the elliptic problem (1 X. F. Chen, Y. W. Qi and M. X. Wang, "Long Time Behavior of Solutions to P-Laplacian Equation with Absorption," SIAM Journal on Mathematical Analysis, p X. F. Chen, Y. W. Qi and M. X. Wang, "Singular Solutions of Parabolic P-Lapacian with Absorption," Transactions of the American MathemaricalSociety, Vol.359  2007, pp.5653-5668.