Multi-dimensional explosion due to a concentrated nonlinear source

Abstract

Let D be a bounded n-dimensional domain, ∂D be its boundary,D be its closure, T be a positive real number, B be an n-dimensional ball {x ∈ D: |x − b| < R} centered at b ∈ D with a radius R, B be its closure, ∂B be its boundary, ν denote the unit inward normal at x ∈ ∂B, and χ B (x) be the characteristic function. This article studies the following multi-dimensional parabolic first initialboundary value problem with a concentrated nonlinear source occupyingB:where the normal vector ν(x) is extended to a vector field defined in the whole domain D, f and ψ are given functions such that f = 0 for x / ∈B, f (0) 0, f (0) 0, f (u) and f (u) are positive for u > 0, and ψ is nontrivial on ∂B, nonnegative, and continuous such that ψ = 0 on ∂D, ψ x i is bounded for i = 1, 2, 3, . . . , n, and ∆ψ + (∂χ B (x)/∂ν)f (ψ(x)) 0 in D. It is shown that it has a unique solution u before a blow-up occurs. A criterion for u to blow up in a finite time is also given. If u exists in a finite time only, then u blows up somewhere onB.  2004 Elsevier Inc. All rights reserved.

The porous medium equation with a concentrated nonlinear source

“…2, B is an n-dimensional ball {x 2 R n : jxj 5 b}, @B is the boundary of B, B (x) is the characteristic function of B, (x) denotes the unit inward normal to x 2 @B, which we extend to a vector field defined in R n , @ B ðxÞ @ denotes the generalized inward normal derivative of B (x), which is a finite normal measure defined in R n . Parabolic equations with concentrated sources were studied by many mathematical workers [1][2][3][4][5][6][7]. A typical concentrated source is a single measure.…”

“…Yuan and Wu [3] considered the Cauchy problem of the porous medium equation with a Dirac measure. Our considerations is motivated by the model by Chan and Tian [5], for the first initial-boundary value problem of the following equation, which corresponds to the special case m ¼ 1 of the Equation (1),…”

“…The aim is to establish the existence of generalized solutions. However, due to m 4 1, namely the appearance of the nonlinear diffusion, makes the theory of Green's function used in [5] inapplicable. In addition, since the source is associated with the nonlinear function f, some results such as the local L 1 estimate of the solutions cannot be obtained by using the methods in [3].…”

A semilinear heat equation with a localized nonlinear source and non-continuous initial data

“…However, as pointed out in , the problem is not well‐defined for . Thus, because the Dirac delta function is the derivative of the Heaviside function, if B is a ball of a natural generalization of the reaction term in is obtained by replacing by the directional derivative , where is the unit inward normal at Considering the latter generalization and a bounded domain Ω, Chan and Tian showed existence and uniqueness of local‐in‐time continuous solutions, as well as, blow‐up properties for the IVP –. Indeed, they proved the existence of an unique local‐in‐time solution u that can blow up only on .…”

A multi-dimensional blow-up problem due to a concentrated nonlinear source in ℝ^{ℕ}

“…We note that such a problem in a bounded domain, instead of R N , was studied by Chan and Tian ( [2], [3]). A solution u is said to blow up at the point (…”