Critical exponents for a semilinear parabolic equation with variable reaction

Abstract. This paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial-boundary value problem of semilinear parabolic equations. The main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution. In the formulation used in this paper, the initial-boundary value problem is transformed into a fixed-point form using an analytic semigroup. The sufficient condition is derived from Banach's fixed-point theorem. This paper also introduces a recursive scheme to extend a time interval in which the validity of the solution can be verified. As an application of this method, the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation.

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“…It is known [6] that for sufficiently large γ, any solution of (24) blows up in finite time. Let h = 2 −4 and τ k = 2…”

Section: Computational Examplementioning

confidence: 99%

A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

Mizuguchi¹,

Takayasu

Kubo

et al. 2017

SIAM J. Numer. Anal.

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“…Hoy en día existen muchos trabajos que tratan sobre a la existencia de soluciones explosivas en problemas de difusión -reacción [2,3,5,6,8,9,10,12,13,15,16,18,19,20,25,27,28,29,30,35,36,41,42,43,44,48,49]. Algunos de estos trabajos discuten la posibilidad de extender las soluciones definidas localmente en [0, T ) a [0, +∞) y que exponen además resultados correspondientes los denominados exponentes de fujita los cuales permiten dar una respuesta apriori sobre la presencia de soluciones explosivas para determinadas ecuaciones de la forma (4) con m = 1ó F (∆u(x, t), x) = ∆u(x, t) en (5).…”

Section: X T) = Div(k(x)∇u(x T)) + F (X T U(x T) ∇U(x T))unclassified

Global existence and blow up of the solution of a problem diﬀusion reaction

Saucedo¹

2017

Sel.mat

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Keywords. diffusion -reaction problem, global existence, local existence, blow up solution, blow up time 1. Introducción. Muchos problemas físicos son modelados analizando el balance de dos fenómenos: la difusión y la reacción. Según [47], puede entenderse al primero como la dispersión de las especies (sustancias) involucradas en el proceso a lo largo del dominio físico del problema (movimiento local), y el segundo, como el proceso de interacción mediante el cual se generan o se consumen las especies involucradas (crecimiento, cambios de estado, etc).Las ecuaciones de difusión -reacción son modelos matemáticos que explican cómo la concentración de una o más especies distribuidas en un espacio (dominio físico) se transforman bajo la influencia de dos procesos: reacciones químicas locales y difusión. Estos tipos de ecuaciones cubren una amplia variedad de modelos físicos en muchos campos de la ciencia, como por ejemplo, procesos biológicos de aparición de manchas en la piel de ciertos animales [30,47], ondas viajantes [23,30,46], dinámica de poblaciones y reacciones químicas [30], entre otros.En general, una ecuación de difusión -reacción es una ecuación en derivadas parciales de segundo orden de la forma:(1) ∂u ∂t (x, t) = div (k(x).∇u(x, t)) + f (x, t, u(x, t), ∇u(x, t)),

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“…This section discusses a Fujita type conclusion for the coupled system (1.1) with variable sources, motivated by the significant result for the scalar case (1.3) (see Theorem 3.7 in [4]). We need two lemmas.…”

Section: Blow-up For Any Initial Datamentioning

confidence: 99%

“…which was studied by Ferreira, de Pablo, Pérez-Llanos and Rossi recently in [4]. The general problem with p(x) ≡ p and q(x) ≡ q in (1.1) was considered by Escobedo and Herrero [3] (see also [6,7] for p, q > 1).…”

Section: Introductionmentioning

confidence: 99%

A semilinear parabolic system with coupling variable exponents

Bai

Zheng

2010

Annali di Matematica

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This paper deals with semilinear parabolic equations coupled via variable sources, subject to the homogeneous Dirichlet condition in a bounded domain. Since the variable exponents in the sources are just assumed to be positive, the non-linearities may be non-Lipschitz. We establish the existence-uniqueness with comparison principle of local solutions to the regularized problem at first, and then consider the maximal solutions of the original problem as the limits of the solutions of the regularized problem. Some criteria are established for distinguishing global and non-global solutions of the problem, dependent or independent of initial data. Especially, we prove a Fujita type conclusion that the solutions blow up for any non-trivial initial data under certain assumptions on the variable sources and the size of the domain.

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Critical exponents for a semilinear parabolic equation with variable reaction

Cited by 51 publications

References 28 publications

A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

Global existence and blow up of the solution of a problem diﬀusion reaction

A semilinear parabolic system with coupling variable exponents

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