The asymptotic behavior of solutions for a class of doubly degenerate nonlinear parabolic equations

AbstractBy using the Morse interaction technique, supposing that the uniqueness of the Barenblatt-type solution is true, the paper studies the large time asymptotic behavior of solutions for the doubly degenerate parabolic equationut=div(|Dum|p−2Dum)−|Dum|p1−uq with initial condition u(x,0)=u0(x). Here the exponents m, p, p1, q satisfy p>p1>p−1, q>m(p−1)>1, p>1, m>1

The well-posedness problem of a hyperbolic–parabolic mixed type equation on an unbounded domain

Abstract(#br)To study the well-posedness problem of a hyperbolic–parabolic mixed type equation, the usual boundary value condition is overdetermined. Since the equation is with strong nonlinearity, the optimal partially boundary value condition can not be expressed by Fichera function. By introducing the weak characteristic function method, a different but reasonable partial boundary value condition is found first time, basing on it, the stability of the entropy solutions is established

Existence and stability of the doubly nonlinear anisotropic parabolic equation

In this paper, we are concerned with a doubly nonlinear anisotropic parabolic equation, in which the diffusion coefficient and the variable exponent depend on the time variable t. Under certain conditions, the existence of weak solution is proved by applying the parabolically regularized method. Based on a partial boundary value condition, the stability of weak solution is also investigated

The weak solutions of a doubly nonlinear parabolic equation related to the p ( x ) p(x)p(x) -Laplacian

Abstract(#br)A nonlinear degenerate parabolic equation related to the p ( x ) $p(x)$ -Laplacian u t = div ( b ( x ) | ∇ a ( u ) | p ( x ) − 2 ∇ a ( u ) ) + ∑ i = 1 N ∂ b i ( u ) ∂ x i + c ( x , t ) − b 0 a ( u ) $${u_{t}}= \operatorname{div} \bigl({b(x)} { \bigl\vert {\nabla a(u)} \bigr\vert ^{p(x) - 2}}\nabla a(u) \bigr)+\sum _{i=1}^{N}\frac{\partial b_{i}(u)}{ \partial x_{i}}+c(x,t) -b_{0}a(u)$$ is considered in this paper, where b ( x ) | x ∈ Ω > 0 $b(x)|_{x\in \varOmega }>0$ , b ( x ) | x ∈ ∂ Ω = 0 $b(x)|_{x \in \partial \varOmega }=0$ , a ( s ) ≥ 0 $a(s)\geq 0$ is a strictly increasing function with a ( 0 ) = 0 $a(0)=0$ , c ( x , t ) ≥ 0 $c(x,t)\geq 0$ and b 0 > 0 $b_{0}>0$ . If ∫ Ω b ( x ) − 1 p − − 1 d x ≤ c $\int _{\varOmega }b(x)^{-\frac{1}{p..

Local Solutions to a Class of Parabolic System Related to the P-Laplacian

Abstract In this paper, the existence and uniqueness of local solutions to the initial and boundary value problem of a class of parabolic system related to the p-Laplacian are studied. The regularization method is used to construct a sequence of approximation solutions, with the help of monotone iteration technique, then we get the existence of solution of a regularized system. By the use of a standard limiting process, the existence of the local solutions of the system is obtained. Finally, the uniqueness of the solution is also proven

Well-posedness problem of an anisotropic parabolic equation

Abstract(#br)In this paper, we are concerned with well-posedness of an anisotropic parabolic equation with the convection term. When some diffusion coefficients are degenerate on the boundary ∂Ω and the others are positive on Ω ‾ , we propose a novel partial boundary value condition to study the stability of the solutions for the anisotropic parabolic equation. A new concept, the general characteristic function of the domain Ω, is introduced and applied. The existence and stability of the solutions is established under the given partial boundary value conditions

On the stability of a non-Newtonian polytropic filtration equation

Abstract(#br)The non-Newtonian polytropic filtration equation with a convection term v t = div ( a ( x ) | v | α | ∇ v | p − 2 ∇ v ) + ∑ i = 1 N ∂ a i ( v , x , t ) ∂ x i $$v_{t}= \operatorname{div} \bigl(a(x) \vert v \vert ^{\alpha }{ \vert {\nabla v} \vert ^{p-2}}\nabla v \bigr)+ \sum_{i=1}^{N}\frac{\partial a_{i}(v,x,t)}{\partial x_{i}}$$ is considered, where p > 1 $p>1$ , α > 0 $\alpha >0$ , a ( x ) ≥ 0 $a(x)\geq 0$ with a ( x ) | x ∈ ∂ Ω = 0 $a(x) | _{x\in \partial \varOmega }=0$ . This kind of equation is degenerate on the boundary, the usual boundary value condition may be overdetermined. Some conditions depending on a ( x ) $a(x)$ and a i ( ⋅ , x , t ) $a_{i}(\cdot ,x,t)$ , which can take place of the boundary value condition, are found. Moreover, how the nonlinear term | v | α..

The Stability of the Solutions for a Porous Medium Equation with a Convection Term

This paper studies the initial-boundary value problem of a porous medium equation with a convection term. If the equation is degenerate on the boundary, then only a partial boundary condition is needed generally. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of the solutions is studied. In some special cases, the stability can be proved without any boundary value condition