MELIK, Ammar2022-05-102022-05-102019-04-23MELIK, Ammar. The global existence of small data solutions of certain viscoelastic evolution problems. 3rd International Conference in Operator Theory, PDE and Applications. April 23-24- 2019. Faculty of Exact Science. Mathematics Department. University of ElOued. [visited in ../../….]. available from [copy the link here]https://dspace.univ-eloued.dz/handle/123456789/11062Working paper.3rd International Conference in Operator Theory, PDE and Applications. April 23-24- 2019. Faculty of Exact Science. Mathematics Department. University of ElOuedWe establish some new results concerning the initial value problem rst order on the whole space Rn (n 1), the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operator, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.englobal existence; small data solutions; certain viscoelastic; evolution problemsWorking Paper