The definition of weak solution and the main results of this paper are listed below.

Definition 2.1

A function (u(x,t)) is said to be a weak solution of equation (1.7) if

$$ begin{aligned} &u in L^{infty}(Q_{T}),qquad u_{t} in L^{1}(Q_{T}), \ &a_{i}(x,t) biglvert u_{x_{i}}^{ frac{alpha _{i}}{2}+1} bigrvert ^{2} in{L^{infty}}bigl(0,T;L^{1}(Omega )bigr),quad i=1,2, ldots , N, end{aligned} $$

(2.1)

and for any function (varphi in C_{0}^{1} ({Q_{T}})), there holds

$$ iint _{Q_{T}} Biggl(-frac{partial varphi}{partial t}u+ sum _{i=1}^{N}a_{i}(x,t)u^{ alpha _{i}}u_{x_{i}} varphi _{x_{i}} Biggr),dx,dt + sum_{i=1}^{N} iint _{{Q_{T}}}b_{i}(u,x,t)varphi _{x_{i}}(x,t),dx,dt =0. $$

(2.2)

The initial value condition is satisfied in the sense of that

$$ lim_{trightarrow 0} int _{Omega } bigl(u(x,t) – u_{0}(x) bigr)phi (x),dx = 0, $$

(2.3)

where (phi (x)in C_{0}^{infty}(Omega )). The boundary value condition (1.9) or the partial boundary value condition (1.12) is satisfied in the sense of trace.

Theorem 2.2

If (alpha _{i}>0), (b_{i}(s,x,t)) is a (C^{1}) function and (vert frac{partial}{partial x_{i}}b_{i}(s,x,t) vert leq c(M)) when (|s|leq M+1), ({u_{0}}(x)) satisfies

$$ u_{0}(x) in L^{infty}(Omega ),qquad a_{i}(x,0)u_{0}^{alpha _{i}} vert u_{0x_{i}} vert in L^{1}(Omega ),quad i=1,2,ldots ,N, $$

(2.4)

(a_{i}(x,t)geq 0) satisfies

$$ iint _{Q_{T}}frac{partial sqrt{a_{i}}}{partial x_{i}},dx,dt leq c,quad i=1,2,ldots ,N, $$

(2.5)

then equation (1.7) with initial boundary values (1.8)(1.9) has a nonnegative solution. Here and the after, M is a constant such that (|u_{0}(x)|_{L^{infty}(Omega)}leq M).

From Theorem 2.3 to Theorem 2.6, we all assume that (a_{i}(x,t)>0), (xin Omega ) and denote that

$$ alpha ^{+}=max {alpha _{1}, alpha _{2}, ldots , alpha _{N}},qquad alpha ^{-}= max {alpha _{1}, alpha _{2}, ldots , alpha _{N}}. $$

Theorem 2.3

Let (u(x,t)) and (v(x,t)) be two solutions of equation (1.7) with the initial value ({u_{0}}(x)), (v_{0}(x)) respectively, and with the boundary value condition (1.9). If (alpha _{i}geq 0), and there is a constant (alpha >frac{1}{2}(alpha ^{+}+2)) such that

$$ biglvert b_{i}(u,x,t)-b_{i}(v,x,t) bigrvert leq c a_{i}(x,t)^{frac{1}{2}} vert u-v vert ^{ alpha},quad i=1,2,ldots ,N, $$

(2.6)

then the solution of equation (1.7) is unique.

Theorem 2.4

Let (u(x,t)) and (v(x,t)) be two nonnegative solutions of equation (1.7) with the initial value ({u_{0}}(x)), (v_{0}(x)) respectively, with the same boundary value condition (1.9). If (alpha _{i}geq 1),

$$begin{aligned}& int _{Omega}a_{i}(x,t)v^{alpha _{i}-1} vert v_{x_{i}} vert ^{2}leq c,qquad int _{Omega}a_{i}(x,t)u^{alpha _{i}-1} vert u_{x_{i}} vert ^{2}leq c,quad i=1,2, ldots ,N, end{aligned}$$

(2.7)

$$begin{aligned}& biglvert b_{i}(u,x,t)-b_{i}(v,x,t) bigrvert leq c a_{i}(x,t)^{frac{1}{2}} vert u-v vert ^{2},quad i=1,2,ldots ,N, end{aligned}$$

(2.8)

then

$$ int _{Omega} biglvert u(x,t)-v(x,t) bigrvert leq c int _{Omega} biglvert u_{0}(x)-v_{0}(x) bigrvert ,dx . $$

(2.9)

Theorem 2.4 implies that we only can show that the stability of weak solutions is true for a kind of solutions which satisfy (2.7). The following stability theorems are established on a partial boundary value condition.

Theorem 2.5

Let (u(x,t)) and (v(x,t)) be two solutions of equation (1.7) satisfying

$$begin{aligned}& frac{1}{lambda} int _{Omega _{lambda t}setminus Omega _{2 lambda t}}a_{i}(x,t) vert u vert ^{alpha _{i}} vert u_{x_{i}} vert ^{2},dx < C(T), end{aligned}$$

(2.10)

$$begin{aligned}& frac{1}{lambda} int _{Omega _{lambda t}setminus Omega _{2 lambda t}}a_{i}(x,t) vert v vert ^{alpha _{i}} vert v_{x_{i}} vert ^{2},dx < C(T), end{aligned}$$

(2.11)

with the initial value ({u_{0}}(x)), (v_{0}(x)) respectively, and with a partial boundary value condition

$$ v(x,t)=u(x,t)=0, quad (x,t)in Sigma . $$

(2.12)

If (alpha ^{-}geq 1), (b_{i}(cdot ,x,t)) satisfies

$$ biglvert b_{i}(u,x,t)-b_{i}(v,x,t) bigrvert leq c sqrt{a_{i}(x,t)} vert u-v vert ,quad i=1,2, ldots ,N, $$

(2.13)

then

$$ int _{Omega} biglvert u(x,t)-v(x,t) bigrvert leq c int _{Omega} biglvert u_{0}(x)-v_{0}(x) bigrvert ,dx . $$

(2.14)

Here,

$$ Sigma = Biggl{ (x,t)in partial Omega times (0,T): sum _{i=1}^{N} sqrt{a_{i}(x,t)} Biggl(prod_{j=1}^{N} a_{j}(x,t) Biggr)_{x_{i}} neq 0 Biggr} $$

(2.15)

and (Omega _{lambda t}= {xin Omega : prod_{i=1}^{N}a_{i}(x,t)> lambda }).

Theorem 2.5 is based on the fact that we can show the first order partial derivative to the solution u is with the local integrability

$$ u_{x_{i}}in L^{infty}bigl(0,T; L^{2}_{mathrm{loc}}(Omega )bigr),quad i=1,2,ldots , N. $$

(2.16)

The weakness of Theorem 2.5 is that the expression of Σ, (2.15) seems too complicated. By choosing another test function, we can prove another stability theorem based on a simpler partial boundary value condition.

Theorem 2.6

Suppose (alpha ^{-}geq 1),

$$ prod_{j=1}^{N} a_{j}(x,t)=0,quad (x,t)in partial Omega times (0,T). $$

(2.17)

Let (u(x,t)) and (v(x,t)) be two solutions of equation (1.7) with the initial value ({u_{0}}(x)), (v_{0}(x)) respectively, but without the boundary value condition. If (2.13) is true and

$$ sum_{i=1}^{N} int _{Omega _{lambda t}setminus Omega _{2lambda t}}a_{i}(x,t) Bigglvert sum _{k=1}^{N}frac{a_{kx_{i}}}{a_{k}} Biggrvert ^{2},dx leq c, $$

(2.18)

then

$$ int _{Omega} biglvert u(x,t)-v(x,t) bigrvert leq c int _{Omega} biglvert u_{0}(x)-v_{0}(x) bigrvert ,dx . $$

We find that the partial boundary value conditions with (2.12) are submanifold of (partial Omega times (0,T)), while in the previous works the corresponding partial boundary value conditions are the cylinder domains (Sigma _{1}times (0,T)), where (Sigma _{1}subseteq partial Omega ) is a relatively open subset [9, 10, 23, 25, 27], etc.

Last but not least, once the well-posedness problem has been solved, we can consider the extinction, blow-up phenomena, the positivity, and the large time behavior of the weak solutions of anisotropic porous medium equation (1.7) in the future. However, different from the porous medium equation, because of the anisotropy, these problems are not so easy to be solved, the methods used in the usual porous medium equation (1.1) cannot be extended to the anisotropic porous medium equation (1.13) directly.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Disclaimer:

This article is autogenerated using RSS feeds and has not been created or edited by OA JF.

Click here for Source link (https://www.springeropen.com/)