The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers’ equation

Deng, Keng; Kwong, Man Kam; Levine, Howard A.

doi:10.1090/qam/1146631

Cited by 45 publications

(21 citation statements)

References 12 publications

(16 reference statements)

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“…We finally consider a comparison principle for positive solutions to (1.2) which will be a fundamental tool in order to prove the existence of global solutions (we refer to [23] and Appendix F in [40] for related comparison theorems for nonlocal problems).…”

Section: Moreover If T < ∞ Then |U(· T)| L ∞ Blows Up As T → T −mentioning

confidence: 99%

Fujita exponents for evolution problems with nonlocal diffusion

García-Melián

Quirós

2009

J. Evol. Equ.

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Section: Moreover If T < ∞ Then |U(· T)| L ∞ Blows Up As T → T −mentioning

confidence: 99%

Fujita exponents for evolution problems with nonlocal diffusion

García-Melián

Quirós

2009

J. Evol. Equ.

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“…In this paper, we shall use a different approach which is motivated by the work of Deng [2]. We also refer the reader to [1,2,3,10,11] for some related works on nonlocal parabolic problems.…”

Section: Stationary Solutionsmentioning

confidence: 99%

A nonlocal quenching problem arising in a micro-electro mechanical system

Guo

Wang

2009

Quart. Appl. Math.

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Abstract. In this paper, we study a nonlocal parabolic problem arising in the study of a micro-electro mechanical system. The nonlocal nonlinearity involved is related to an integral over the spatial domain. We first give the structure of stationary solutions. Then we derive the convergence of a global (in time) solution to the maximal solution as the time tends to infinity. Finally, we provide some quenching criteria.

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“…, with 1 r , was studied in [DKL,D,S1] in a bounded domain. In particular, in the case r= , interesting results were obtained in [D] concerning the influence of the values of p and q on the size of the blow-up set.…”

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confidence: 99%

Uniform Blow-Up Profiles and Boundary Behavior for Diffusion Equations with Nonlocal Nonlinear Source

Souplet

1999

Journal of Differential Equations

204

138

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In this paper, we introduce a new method for investigating the rate and profile of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solutions have global blowup and that the rate of blow-up is uniform in all compact subsets of the domain. This results in a flat blow-up profile, except for a boundary layer, whose thickness vanishes as t approaches the blow-up time T*. In each case, the blow-up rate of |u(t)| is precisely determined. Furthermore, in many cases, we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer. The size of the boundary layer then decays like -T*&t, and the solution u(t, x) behaves like |u(t)| d(x)Â-T *&t in the boundary layer, where d is the distance to the boundary. Some Fujita-type critical exponents results are also given for the Cauchy problem. Academic Press

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The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers’ equation

Cited by 45 publications

References 12 publications

Fujita exponents for evolution problems with nonlocal diffusion

Fujita exponents for evolution problems with nonlocal diffusion

A nonlocal quenching problem arising in a micro-electro mechanical system

Uniform Blow-Up Profiles and Boundary Behavior for Diffusion Equations with Nonlocal Nonlinear Source

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