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Global Well posedness for Fractional Sobolev Galpern Type Equations
Global Well posedness for Fractional Sobolev Galpern Type Equations
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Date
2021
Authors
Huy Tuan Nguyen
Nguyen Anh Tuan
Chao Yang
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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.
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Keywords
"fractional pseudo-parabolic,
globally Lipschitz source,
exponential nonlinearity,
global well-posedness."