Exponential growth for a semilinear viscoelastic heat equation with Lpp (Rn)-norm in bi-Laplacian type

Abstract

The problem considered here is a class of semi-linear visco-elastic heat equations in bi-Laplacian type. We introduce a weighted space to overcome the difficulties in the non-compactness of some operators and some useful Sobolev embedding inequalities. Under certain conditions on the parameters p, ρ, ɳ, we prove that the local solutions grow as an exponential function in the L_p^p-norm, i.e. ||u||^p_Lp^p(Rn) →+∞ as t tends to +∞.