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In this paper, the existence and the uniqueness of an evolutionary anisotropic $ p_i(x) $ p i (x) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the $ L^{\infty } $ L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the $ L^{\infty } $ L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic $ p(x)- $ p (x) − Laplace equation with convection term via $ L^{\infty }- $ L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc. [ABSTRACT FROM AUTHOR]

In this paper, we consider anisotropic parabolic equations with variable exponents. The existence of weak solution is proved by means of the parabolically regularized method, and the stability of nonnegative solutions is analyzed under certain conditions. By introducing a new kind of weak solution, the stability of the weak solution is established without any boundary value condition. [ABSTRACT FROM AUTHOR]

Abstract The strong degenerate parabolic equation ∂xxu+u∂yu−∂tu=f(x,y,t,u) $$ \partial _{xx}u+u\partial _{y}u-\partial _{t}u =f(x,y,t,u) $$ comes from the mathematics of finance. A kind of entropy solution is introduced. By Kružkov’s bi-variable method, the stability for entropy solutions only depending on the initial value is proved. The usual boundary value condition is replaced by the regularity of the domain in a special sense.

The nonlinear diffusion equation of the ideal barotropic gas through a porous medium is considered. If the diffusion coefficient is degenerate on the boundary, then the solutions may be controlled by the initial value completely, the well-posedness of the solutions may be obtained without any boundary condition.

Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

For the stability of degenerate parabolic equations, the usual boundary value condition may be overdetermined. How to find an optimal partial boundary value condition has been a long-standing problem. In this paper, we develop the weak characteristic function method to establish a reasonable partial boundary value condition, no matter whether the spatial variable domain Ω is bounded or unbounded. By means of the Kruzkov bi-variables method, the stability of entropy solutions dependent on this partial boundary value condition is proved. [ABSTRACT FROM AUTHOR]