Qualitative Research On Semilinear Pseudo-parabolic Equations And Reaction-diffusion Equations At High Energy Level

In the first part of this thesis, we study a class of semilinear pseudo-parabolic equations.Pseudo-parabolic equation describes a variety of important physical processes, such as the ag-gregation of population and the unidirectional propagation of nonlinear, dispersive, long waves.In previous studies, the authors proved global existence exponent and the critical Fujita ex-ponent for a class of semilinear pseudo-parabolic equations by integral method and contrac-tion mapping principle. The authors proved the finite time blow up of solutions for semi-linear pseudo-parabolic equations with p∈（1,1+2/n） and nontrivial initial data. However,p∈（1,1+2/n） and the interval （1,1+2/n） is very small for sufficiently large n. In other words,there are no results on global well-posedness of solutions for semilinear pseudo-parabolic equa-tions under the condition p∈（1,+∞）. Furthermore, the authors only proved the finite timeblow up of solutions with sufficiently large initial data and p∈（1+2/n,+∞）, but they didnot point out how large the initial data is. It is still open that whether the global solutions ex-ist when the initial data is not sufficiently large. In order to resolve the open problems, wediscuss global existence and finite time blow up of solutions for a class of semilinear pseudo-parabolic equations. By introducing a family of potential wells, we first prove the invarianceof solutions in some sets, and obtain global existence, nonexistence and asymptotic behaviorof solutions with initial energy J（u₀）≤d. Moreover, we use comparison principle and varia-tional methods to solve semilinear pseudo-parabolic equations with strong damping term, andobtain global solutions and finite time blow up solutions with high initial energy J（u₀）> d. Wereveal the influence of strong damping term and nonlinear source terms on the well-posednessof semilinear pseudo-parabolic equations by assuming the restrictions for the initial data andexponents of nonlinear terms. Finally, we enrich and develop the theory system of semilinearpseudo-parabolic equations.In the second part of this thesis, we study a class of reaction-diffusion equations with sev-eral nonlinear source terms of different signs. Reaction-diffusion equations have been proposedin the study of biological invasion and disease spread. For a class of reaction-diffusion equationswith several nonlinear source terms of different signs, the authors obtained the global existenceand finite time blow up of solutions with J（u₀）≤d in previous studies. The depth of potentialwell d for reaction-diffusion equations is usually very small, which means mostly the initial conditions u₀don’t satisfy the high energy case such as J（u₀）> d. Using the potential wellmethod to analyze finite time blow up of solutions with J（u₀）≤d is a small part of functionalspace. In order to remove the energy limit, we analyze the behavior of the solutions when theinitial data varies in the phase space H₀¹（Ω） by the comparison principle and variational meth-ods. Finally, we obtain finite time blow up of solutions with high initial energy J（u₀）> d,which perfect the study on a class of reaction-diffusion equations with several nonlinear sourceterms of different signs.