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In this paper, we deal with a p(x)-biharmonic heat equation with variable exponent under Dirichlet boundary and initial condition. We prove the blow up of solutions under suitable conditions.

Our aim here is to collect and to compare two definitions of the fractional powers of non-negative operators that can be found in the literature; we will present the proof of an equivalence and compare properties of that notions in different approaches. Then we will apply next this equivalence in the study of global and local existence of solution for the semilinear Cauchy problem in R N with fractional Laplacian ut = -(-∆)au + f(x, u), u(0, x) = u0(x), x ∈ R N .

In this paper, we consider the following Timoshenko equation utt + ( 2u ( M ((( ( u (( 2( ( u + ut = (u( q-1 u associated with initial and Dirichlet boundary conditions. We prove the non-existence of solutions with positive and negative initial energy.

In this paper, we consider a nonlinear beam equation with a strong damping and the p(x)-biharmonic operator. The exponent p(·) of nonlinearity is a given function satisfying some condition to be specified. Using Faedo-Galerkin method, the local and global existence of weak solutions is established with mild assumptions on the variable exponent p(·). This work improves and extends many other results in the literature. [ABSTRACT FROM AUTHOR]

Our aim here is to collect and to compare two definitions of the fractional powers of non-negative operators that can be found in the literature; we will present the proof of an equivalence and compare properties of that notions in different approaches. Then we will apply next this equivalence in the study of global and local existence of solution for the semilinear Cauchy problem in RN with fractional Laplacian.... [ABSTRACT FROM AUTHOR]

A Kirchhoff equation type with memory term competing with a logarithmic source is considered. By using potential well theory, we obtained the global existence of solution for the initial data in a stability set created from Nehari Manifold and prove blow up results for initial data in the instability set. [ABSTRACT FROM AUTHOR]