Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Arrieta, José M.; Rodriguez-Bernal, Anı́bal

doi:10.1081/pde-200033760

Cited by 24 publications

(20 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…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

See 1 more Smart Citation

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

Saucedo¹

2017

Sel.mat

View full text Add to dashboard Cite

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)),

show abstract

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

“…En [2,3,8,13] estudian problemas como (3) y (4). Se establecieron resultados complementarios sobre existencia global de soluciones y explosión de las mismas.…”

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

View full text Add to dashboard Cite

show abstract

“…We suspect that the φ n 's, modulo rotations, represent all solutions to the nonlinear boundary value problem with Neumann data λ sinh(u) on B 1 . In what follows we will consider solutions u(x, y, t) to…”

Section: A Two-space-dimensional Casementioning

confidence: 99%

Transient behavior of solutions to a class of nonlinear boundary value problems

Bryan

Vogelius

2011

Quart. Appl. Math.

View full text Add to dashboard Cite

Abstract. In this paper we consider the asymptotic behavior in time of solutions to the heat equation with nonlinear Neumann boundary conditions of the form ∂u/∂n = F (u), where F is a function that grows superlinearly. Solutions frequently exist for only a finite time before "blowing up." In particular, it is well known that solutions with initial data of one sign must blow up in finite time, but the situation for sign-changing initial data is less well understood. We examine in detail conditions under which solutions with sign-changing initial data (and certain symmetries) must blow up, and also conditions under which solutions actually decay to zero. We carry out this analysis in one space dimension for a rather general F , while in two space dimensions we confine our analysis to the unit disk and F of a special form.

show abstract

“…Hence, nowadays the underlying mechanisms for dissipativeness or blow-up of solutions is fairly well understood; see e.g. [7,4,6,18,19]. Therefore, it is a natural question to analyse the dynamics and bifurcations induced by the nonlinear boundary conditions, and compare its effects with the case of an interior reaction term, which has been more widely studied.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity

Arrieta¹,

Pardo²,

Rodriguez-Bernal³

2007

Proceedings of the Royal Society of Edinburgh: Section A Mathem

Self Cite

View full text Add to dashboard Cite

We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, characterize when they are sub- or supercritical and analyse the stability type of the solutions. Furthermore, we apply these results and techniques to obtain Landesman–Lazer-type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti-maximum principle.

show abstract

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Cited by 24 publications

References 15 publications

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

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

Transient behavior of solutions to a class of nonlinear boundary value problems

Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity

Contact Info

Product

Resources

About