Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity

Abstract

Keywords:Variable exponent Lebesgue and Sobolev spaces Nonlinear parabolic variational inequality In this article, new properties of variable exponent Lebesgue and Sobolev spaces are examined. Using these properties we prove the existence of the solution of some parabolic variational inequality.

Evolutionary Variational Inequalities with Volterra Type Operators

“…Let us remark that problems for evolutionary variational inequalities without Volterra type operators are widely investigated, particularly in [5], [8], [9], [15], [16], [23], [24], [25], [26], and others. Most of the results, which are known to us, were obtained by using the Banach's fixed point theorem (the principle of compression mappings).…”

Initial-boundary-value problems for anisotropic elliptic-parabolicpseudoparabolic equations with variable exponents of nonlinearity

“…The spaces were first introduced in [16]. The properties of these spaces and their applications to nonlinear differential equations with variable exponents of nonlinearity have been actively studied (see, e.g., [17][18][19][20][21][22][23][24]). But we do not known works where to consider the anisotropic elliptic-parabolic-pseudoparabolic equations with variable exponents of nonlinearity.…”

On global existence and bounds for the blow-up time in a semilinear heat equation involving parametric variable sources

“…From a mathematical point of view, equations of the types (1) and ( 2) with variable exponent are usually referred to as equations with nonstandard growth conditions. Under certain conditions on the initial data and certain ranges of exponents, the existence, uniqueness and other qualitative properties of solutions for parabolic and hyperbolic equations with variable nonlinearity have been studied by many authors (see [9][10][11][12][13][14][15][16][17] and references therein).…”