Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory

<p style='text-indent:20px;'>In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.</p>

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Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption

Journal of Applied Mathematics ◽

10.1155/2012/648067 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17

Author(s):

Yulan Wang ◽

Zhaoyin Xiang ◽

Jinsong Hu

Keyword(s):

Diffusion Equation ◽

Existence Of Solutions ◽

Neumann Boundary ◽

Reaction Diffusion ◽

Nonlocal Diffusion ◽

Reaction Diffusion Equation ◽

Optimal Conditions ◽

All Solutions ◽

Blowing Up ◽

Nonlocal Reaction

We investigate a nonlocal reaction diffusion equation with absorption under Neumann boundary. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of solutions blowing up in finite time, or for the global existence and boundedness of all solutions. For the blowup solutions, we also study the blowup rate estimates and the localization of blowup set. Moreover, we show some numerical experiments which illustrate our results.

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Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

Open Mathematics ◽

10.1515/math-2020-0088 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1552-1564

Author(s):

Huimin Tian ◽

Lingling Zhang

Keyword(s):

Boundary Conditions ◽

Blow Up ◽

Sufficient Conditions ◽

Neumann Boundary ◽

Reaction Diffusion ◽

Neumann Boundary Conditions ◽

Reaction Diffusion Equations ◽

Time Dependent ◽

Diffusion Equations ◽

Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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Reduced basis approximations of the solutions to spectral fractional diffusion problems

Journal of Numerical Mathematics ◽

10.1515/jnma-2019-0053 ◽

2020 ◽

Vol 28 (3) ◽

pp. 147-160

Author(s):

Andrea Bonito ◽

Diane Guignard ◽

Ashley R. Zhang

Keyword(s):

Computational Cost ◽

Fractional Power ◽

Reaction Diffusion ◽

Fractional Diffusion ◽

Diffusion Problem ◽

Quadrature Point ◽

Reduced Basis ◽

Fast Evaluation ◽

Bottle Neck ◽

Diffusion Problems

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

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