The Study Of Solutions To Several Evolution Equations (Systems) Involving The Fractional Laplacian

The fractional Laplacian is a type of nonlocal pseudo-differential operator,which is called a fractional diffusion flux and often appears in several remote or anomalous physical phenomena,such as a L?evy flight,disturbance,or anomalous diffusion of plasma.Compared with the integer-order operators,the fractional Laplacians are more accurate in describing some of the physical phenomena and dynamic processes.Therefore,an increasing number of studies have been devoted to the study of fractional differential equations.In particular,the nonexistence of nontrivial global solutions and the existence of nonnegative solutions have been extensively researched by mathematicians and numerous results of the related problems have been proved.In this paper,we mainly study the nonexistence of nontrivial global solutions for several classes of evolution equations(systems)involving the fractional Laplacian and the existence of nonnegative solutions for semilinear pseudo-parabolic inequality.Using the test function method,we prove the nonexistence of nontrivial global solutions for the considered equations(systems),and derive the blow-up conditions of the solutions.First,establishing the definitions of the weak solutions for the equations(systems).Then,we obtain the integral estimates of the definitions of the weak solutions by utilizing H?lder's inequality,?-Young's inequality and Ju's inequality,etc.Moreover,combining the properties of the test function and the definitions of the fractional Laplacian,we obtain the estimates of the inequalities separately.Finally,according to the estimates,we obtain the nonexistence of nontrivial global solutions,and derive the blow-up conditions of the solutions.In addition,based on the method of supersolution and the modified Bessel function,we consider the existence of nonnegative solution for the semilinear pseudo-parabolic inequality.Because the equations(systems)have no restrictive conditions regarding the initial values,the definitions of the weak solutions are obviously different from the definitions in the existing literatures.However,owing to the nonlocality of the fractional Laplacian and the appearance of the pseudo-parabolic third-order term,the proofs of the pseudo-parabolic equations(systems)are more complicated than that for the corresponding parabolic equations(systems).The proofs of nonexistence are based on the suitable choice of the test functions,this is the key to prove the nonexistence in this paper.The existence result is established by the construction of an explicit nonnegative solution of the semilinear pseudo-parabolic inequality.