Existence Of Solutions For Several Nonlinear Time-space Fractional Evolution Equations

Nonlinear evolution equations have important applications in many fields,such as mathematics,physics,chemistry,biology,engineering,materials and so on.In recent years,fractional derivatives can better describe some natural phenomena,such as memory effect,anomalous diffusion and Lévy processes,therefore nonlinear fractional partial differential equations have attracted a great deal of attention.However,different from integral derivatives,Caputo time fractional derivatives and fractional Laplace operators both have non-locality and singularity,thus many classical theories and methods are no longer fully applicable,which brings some difficulties to the study of related problems.Especially,the appearance of fractional derivatives and nonlinear terms further increases lots of difficulties.At present,most studies focus on nonlinear time or space fractional evolution equations,but there are relatively few results on the existence of solutions of nonlinear time-space fractional evolution equations,which arouses strong research interest.In this dissertation,we study the existence,uniqueness,asymptotic behavior,blowup and regularity of solutions for nonlinear evolution equations with Caputo time fractional derivatives and fractional Laplace operators.The main contents are as follows:Firstly,we study existence,uniqueness and decay properties of global weak solutions for time-space fractional Kirchhoff-type diffusion equations,where the nonlinear source term corresponds to two different cases.By the spectral theory of fractional Laplace operators,Gal?rkin method and iterative techniques,we obtain global existence and uniqueness of the approximating solution.By properties of Caputo fractional derivatives and fractional Sobolev spaces,we generalize classical compactness theorem to the time fractional case,and get some prior estimates of weak solutions respectively.Then,we obtain existence,uniqueness and decay estimates of the global weak solution.Secondly,we investigate global existence,uniqueness and blow-up of mild solutions for abstract time-space fractional diffusion equations.By analytical techniques,we give some assumptions of the nonlinear term,then we discuss existence and nonexistence of global mild solutions under different cases.For the subcritical case,by properties of timespace fractional derivatives,we generalize classical strong maximum principle to the timespace fractional case.Further,by upper and lower method,we obtain global existence,uniqueness and asymptotic behavior of the mild solution.For critical and supercritical cases,we prove local existence and uniqueness of the mild solution by contraction mapping principle.Further,by constructing suitable assumptions and test functions,we prove that mild solutions blow up in finite time.Thirdly,we study existence,uniqueness,asymptotic behavior and regularity of global weak solutions for time-space fractional Rosenau equations,where the nonlinear term involves fractional Laplace operators.By Laplace transforms and Mittag-Leffler functions,we obtain explicit expressions of the solution operator and some rigorous estimates,further we get existence,uniqueness and asymptotic behavior of global weak solutions for the corresponding linear problem.By precise calculations,we construct a class of timeweighted fractional Sobolev spaces.Then,we obtain existence,uniqueness and asymptotic behavior of the global weak solution by contraction mapping principle.Further,we discuss regularity of global weak solutions.Finally,we investigate global existence,uniqueness and blow-up of strong solutions for time-space fractional porous media equations with nonlocal reaction terms.By subdifferential approaches,we first establish existence,uniqueness and decay estimates of global strong solutions for the corresponding problem with local reaction terms.Further,by accurate calculations,we get a critical exponent for the equation with nonlocal reaction terms,then we discuss existence and nonexistence of global strong solutions under different cases.For critical and supercritical cases,by time-space fractional derivatives,contraction mapping principle and inequality techniques,we obtain existence and uniqueness of the global strong solution.For the subcritical case,by constructing suitable assumptions and test functions,we prove that strong solutions blow up in finite time.