Study of the solutions to a model porous medium equation with variable exponent of nonlinearity

Abstract

The authors of this paper study the Dirichlet problem of the following equationThe existence and uniqueness of weak solutions are proven. Also, the properties of the solutions are studied which include the property of finite speed of propagation of disturbances, localization property and the property of vanishing at a finite time etc.

Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents

“…Quasi-linear parabolic equations with variable exponents appear in physical problems like electrorheological fluids [5,10,11], image processing [1,4,7] and porous medium equations [2,12]. A representative example is the system…”

Blow‐up of solutions for a viscoelastic wave equation with variable exponents

“…In this section, we show the local existence of solutions for system (1)- (3). For this purpose, we consider a related initial-boundary value problem.…”

“…However, to our knowledge, there is no blow-up result of solutions for the viscoelastic hyperbolic equations with variable exponents. We prove a finite time blow-up result of solutions with positive initial energy for the problem (1)- (3).…”

“…More details on these problems can be found in previous studies. () In fact, there are only few works regarding hyperbolic equations with variable exponents of nonlinearity. Messaoudi et al showed the existence and blow‐up of solutions for the nonlinear damped wave equation with variable exponents: where a , b are positive constants and the exponents m (·) and p (·) are given measurable functions.…”

“…More details on these problems can be found in previous studies. [1][2][3] In fact, there are only few works regarding hyperbolic equations with variable exponents of nonlinearity. Messaoudi et al 4 showed the existence and blow-up of solutions for the nonlinear damped wave equation with variable exponents:…”