On a nonlocal degenerate parabolic problem (original) (raw)
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Proceedings of the American Mathematical Society, 2000
We establish existence and uniqueness of solutions for a general class of nonlocal nonlinear evolution equations. An application of this theory to a class of nonlinear reaction-diffusion problems that arise in population dynamics is presented. Furthermore, conditions on the initial population density for this class of problems that result in finite time extinction or persistence of the population is discussed. Numerical evidence corroborating our theoretical results is given.
On a class of parabolic equations with nonlocal boundary conditions*1
Journal of Mathematical Analysis and Applications, 2004
In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.
On a class of parabolic equations with nonlocal boundary conditions
Journal of Mathematical Analysis and Applications, 2004
In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.
A nonlinear degenerate parabolic equation
Annali della Scuola Normale Superiore di Pisa Classe di Scienza, 1977
The author deals with the equation u_t = (a(u) u_x)_x + b(u) u_x in which subscripts denote partial differentiation. The Cauchy problem, the Cauchy-Dirichlet problem in a semi-infinite strip, and the Cauchy-Dirichlel problem in a rectangle are studied. Weak solutions are formulated. The main results are existence, uniqueness, and local regularity theorems. Further it is shown how it is possible to extract maximum principles for weak solutions of the problems from existence proofs. The author also presents necessary and sufficient conditions for weak solutions of the problems to vanish in an open subset of their domain of definition.