Blow-up Properties In The Parabolic Problems With Anisotropic Nonstandard Growth Conditions

In this thesis,we study the homogeneous Dirichlet initial-boundary problem for a class of nonlinear parabolic partial differential equation with anisotropic variable exponents,where the nonlinearity comes from the product of the loc al and nonlocal inner source.Such parabolic problem with variable exponents reflects the point-wise physical phenomenon,and is widely used in electro-rheological fluid,digital image processing model etc.We use the basic theory of parabolic equations and variable Sobolev space to show the blow-up versus global existence results of solutions by constructing the energy function and control function,and using the comparison principle.We first give the existence and uniqueness of weak solutions in some variable Sobolev space.Then,for the case that the maximum value of the variable exponent is greater than 1,we use the modified Kaplan method to give the blow-up criterion.Under the condition that the maximum value of the variable exponent is less than 1,based on the condition of the parabolic partial differential equation with variable exponents,the blow-up criteria of the solution are obtained for the positive and negative energy initial conditions respectively.After that,we use the comparison principle of the variable exponent problem and construct suitable upper solution to show the global existence of the solution.At last,we obtain the asymptotic properties of the blow-up solutions,including blow-up time estimate and blow-up rate.It is interesting that variable coefficients and variable exponents influent the blow-up properties sensitively.We also find that the variable exponent and coefficient play an important role in the Fujita blow-up phenomena of the parabolic equation with variable exponent.