ON THE LONG TIME BEHAVIOR OF A VISCOELASTIC PLATE EQUATION WITH A VELOCITY-DEPENDENT MATERIAL DENSITY AND A LOGARITHMIC NONLINEARITY

In this paper, we are concerned with the following problem |u_t|^ρu_tt + ∆²u + ∆²u_tt − ∫_t h(t − s)∆²u(s)ds = αu ln |u|. We use the multiplier method, some logarithmic inequalities and some properties of integro-differential inequalities to establish a general decay result for the solution of this problem. We minimize the conditions imposed on the relaxation function h by assuming that h satisfies h^′ (t) ≤ −ξ(t)H(h(t)), where the two functions ξ and H satisfy some conditions. This assumption allows us to use a more general class of the relaxation functions and to obtain a more general stability result. In fact, our results generalize, extend and improve many results in the literature.