Global Existence And Blow-Up Of Solutions For Two Kinds Of Space-Time Fractional Pseudo-Parabolic Equations

Fractional differential equations play an important role in describing abnormal diffusion and dynamic systems with chaotic dynamics in practical applications.As a non-classical diffusion equation,the pseudo-parabolic equation has a rich physical background,such as the theory of percolation through a fractured and delayed uniform liquid.Many important physical models and practical problems require us to consider pseudo-parabolic models with fractional derivatives,such as those of viscoelastic fluids.Therefore,the study of space-time fractional pseudo-parabolic equation has important theoretical and practical significance.In this paper,the global existence and blow-up of solutions of two kinds of space-time fractional pseudo-parabolic equations are studied.The first chapter introduces the research background,main results and basic knowledge.In Chapter 2,we consider the blow-up and global existence of Cauchy problems for a class of space-time fractional pseudo-parabolic equations.A family of solution operators is defined based on a kind of density function and semigroup,and the Lp-Lq estimate of corresponding linear problem solutions is investigated.On this basis,the local existence of solutions to a class of space-time fractional pseudo-parabolic equations is studied by using the fixed-point theorem.The definition of weak solutions is given and it is proved that mild solutions are also weak solutions.The global existence of solutions is proved by using the contraction mapping principle,and the blow-up of solutions is proved by using the test function method.In Chapter 3,we discuss the blow-up and global existence of a class of fractional pseudoparabolic equations with nonlinear memory terms in a bounded domain.Firstly,the solution operator is defined by a kind of density function and semigroup,and its properties are investigated.According to these properties,we obtain the local existence of mild solutions by using the contraction mapping principle,and prove that mild solutions are also weak solutions.Secondly,the global existence of the solution is proved by using the contraction mapping principle.Finally,the blow-up of fractional pseudo-parabolic equations is proved by contradiction.The obtained Fujita critical exponents are consistent with the Fujita critical exponents of the time-fractional differential equations with nonlinear memory terms,and the diffusion effect of the corresponding higher order terms is not sufficient to affect the Fujita critical exponents.