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Advances in Studying Weakly Coupled Nonlinear Parabolic Systems

Author: LiuXiangYang
Tutor: GaoWenJie
School: Jilin University
Course: Applied Mathematics
Keywords: Nonlinear Existence Stability Blow-up Parabolic
CLC: O175.26
Type: Master's thesis
Year: 2010
Downloads: 30
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Abstract


With the continuous study of partial differential equations, the nonlinear parabolic equa-tions have been applied extensively in practical problems, the research of nonlinear parabolic equations abstracted from the practial problems has drawn increasing concern. And there have been many works about the parabolic equations, most of which studied mathematical models of the nonlinear parabolic equations abstracted from physical, chemical and ecolog-ical phenomena and so on, such as the semiconductor model and combustion model and so on. Widely reading papers, I learned the properties of the solutions of the nonlinear parabolic equations, such as the existence, stability, and blow-up, in different actual background. From the context the nonlinear systems owning different background, the systems have certain ex-tensiveness. So the nonlinear weakly coupled parabolic equations are described in this paper.In the first part, the existence of the solutions is introduced.First, a class of quasi-linear parabolic equations with homogeneous Neumann boundary conditions in the bounded domain are introduced. Many authors studied the existence and uniqueness of the local solutions. Then the global existence of classical solution of the parabolic equations is given by the maximum principle and comparison principle.Next, we introduce the existence of solutions of a class of degenerate quasi-linear parabolic equations with nonlocal source. In this section we mainly introduce the existence of nonnegative solutions and the global existence of solutions of the equations in special case, and also the existence of solution of a similar kindled models with nonlocal source. The global existence of solutions of the system is studied mainly by the comparison princi-ple and the method of subsolution and supersolution.Finally, we introduce the global finiteness of the solutions of a evolution p-Laplace systems with nonlinear boundary conditions. In this section we introduce the concept of ε-subsolution andε-supersolution and prove the existence of positive solution by the com-parison principle and the method of subsolution and supersolution.In the second part, the stability of solutions of the nonlinear parabolic equations is introduced.This part begins considering the stability of solutions of the following weakly-coupled parablic system. hereΩis a bounded domain in R", and the boundary u(x, t) is a vector-valued function with the component is the gradient of function ue(x, t) with respect to the space variable. The functon aijθ, aθ, gθis continuously differentiable with respect to their components. Dividing the existing space of the stable solutions into three parts independently, the authors considered the stability and instability of the stable solutions.Then we present the stabilization of the minimal nonnegative solution of the following system of semilinear parabolic equations with the initial conditions where k is a positive integer,In the third part, the blow-up of the solutions is introduced. AbstractThe solutions of the nonlinear parabolic equations will blow-up in finite time. It is one of the reason that we are interested in nonlinear parabolic equations. Consider the following nonlinear parabolic equations whereΩ(?) RN is a bounded domain, and (?)Ω is smooth. xo is a fixed point inΩ. p1,q1, p2, q2 are nonnegative constants. We mainly consider the blow-up of the system, also the relation-ship between the blow-up of the solutions in finite time and the exponents in the equations.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Parabolic equation
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