| There are a lot of nonlinear problems depending on continuous time, which came from physics chemistry, fluid mechanics, biology and other science. In mathematics, these problems can be modeled by nonlinear evolution equation with singularity or degeneracy. In this thesis, we mainly analyze the singularity of solutions to several nonlinear evolution equations arose in applied sciences. We divide the dissertation into six chapters:In Chapter1, we first introduce the background and development of the problems in this thesis, and then state our main results.In Chapter2, we consider the Cauchy problem for a higher order shallow water equation, yt+auxy+buyx=0, where y:=A2ku=(1-(?)x2)ku. The local well-posedness of solutions for the Cauchy problem in Sobolev space Hs(R) with s>7/2is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are also acquired. Finally, the weak solution for the equation is considered.(The main results of this chapter are published in J. Differential Equations,2011(251):3488-3499)In Chapter3, we devote to the continuation of solutions to the generalized Camassa-Holm equation beyond wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear system. This formulation allows one to continue the solution after collision time, giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global conservative or dissipative solutions, which depend continuously on the initial data.(The main results of this chapter are accepted in Discrete Contin. Dyn. Syst.)In Chapter4, we deal with the Cauchy problem for a weakly dissipative shallow water equation with high-order nonlinearities yt+um+1yx+bumuxy+λy=0, where A, b are constants and m(?)N, the notation y:=(1-(?)x2)u, which includes the famous Camassa-Holm, Degasperis-Procesi and Novikov equations as special cases. The local well-posedness of solutions for the Cauchy problem in Besov space Bp,rs(R) with1≤p,r≤+∞and s> max{1+1/p,3/2} is obtained. Under some assumptions, the existence and uniqueness of the global strong solutions and conditions that lead to the development of singularities in finite time for the solutions are acquired. The analyticity and persistence properties of strong solutions for the equation are also considered. Furthermore, although the H1(R)-norm of the solution to the nonlinear model does not remain constants, the existence of its weak solutions in low order Sobolev space HNs(R) with1<s<3/2is established under the assumptions λ=0, u0(x)(?)HN(R) and‖u0nu0n‖L∞(R)<∞. Finally, the global weak solution and peakon solution for the equation with λ=0are also given.(The main results of this chapter are submitted to Nonlinearity)In Chapter5, we devote to investigate the quenching phenomenon for a reaction-diffusion system with coupled singular absorption terms. The solution of the system quenches in finite time for any initial data is obtained, and the blow-up of time-derivatives at the quenching point is verified. Moreover, under appropriate hypotheses, the criterions to identify the simultaneous and non-simultaneous quenching are found, and the four kinds of quenching rates for different nonlinear exponent regions are given. Finally, some numerical experiments are performed, which illustrate our results.(The main results of this chapter are published in Boundary Value Problems,2010, Article ID797182.)In Chapter6, we study the quenching phenomenon for a reaction-diffusion equation with nonlinear memory subject to positive Dirichlet boundary condition. The local existence and uniqueness of the solution are proved, moreover, there exists a critical length α*such that the solution quenches in finite time for α>α*, and the blow-up of time-derivatives at the quenching point is verified. Under appropriate hypotheses, the quenching rate estimates are given. Finally, some numerical experiments are performed, which illustrate our results.(The main results of this chapter are published in Commun. Nonlinear Sci. Numer. Simulat.,2012(17):754-763.)In Chapter7, we deal with the quenching phenomenon for a non-local diffusion equation with a general singular absorption term and Neumann boundary condition. The local existence and uniqueness of the solution are proved, and the solution of the equation quenches in finite time is shown. Moreover, under appropriate condition, the only quenching point is x=0and the estimates of the quenching rate are obtained. Finally, some numerical experiments are performed, which illustrate our results.(The main results of this chapter are published in Z. Angew. Math. Phys.,2011(62):483-493.)... |
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