Global well-posedness for fractional Sobolev-Galpern type equations

Abstract

This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally use of Orlicz space.

Novel spectral schemes to fractional problems with nonsmooth solutions

“…Theoretical and practical investigations regarding this branch have been performed. Some important theoretical results can be found in [1][2][3][4][5][6][7]. Many models that describe many phenomena in the applied sciences can be modeled by fractional differential equations (FDEs); see, for example, [8][9][10][11][12].…”