Qualitative study of certain evolution problems

Abstract

In this thesis, we are interested in the study of the existence and uniqueness of global solutions, as well as, the blow up in finite time of solutions for a certain systems of semilinear Volterra integro differential equations of parabolic and hyperbolic type. Especially, the non-linear part is defined by an integral terms over the past history of the nonlinear forcing containing fractional time-dependent convolution kernels. We study this type of generalized problems to obtain similar results to those obtained in the case of an equation. We will see that under certain conditions on the exponents, the order of the temporal fractional derivatives there is a critical value of the dimension space for which the global with small data solution results as well as the explosion in finite time with initial conditions having positive average are obtained. The methodology to be followed to demonstrate the global existence and the asymptotic behavior based essentially on the use of the semi-group method combined with a priori estimates in the Lebesgue spaces. In parallel, in the study of the blow-up in finite time result, we will focus on the concept of weak solutions and its connection with the mild ones and thus via the test functions method's get the desired results.