Influence of the Hardy potential in a semilinear heat equation

2009 ◽  
Vol 139 (5) ◽  
pp. 897-926 ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Ireneo Peral ◽  
Ana Primo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Critical Power ◽  
Regular Domain ◽  
Hardy Potential ◽  
Semilinear Heat Equation ◽  
The Cauchy Problem ◽  
Image Position ◽  
Class Of Functions ◽  
Bounded Regular Domain

This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problemwhere Ω⊂ℝN, N≥3, is a bounded regular domain such that 0∈Ω, p>1, and u0≥0, f≥0 are in a suitable class of functions.There is a great difference between this result and the heat equation (λ=0); indeed, if λ>0, there exists a critical exponent p+(λ) such that for p≥p+(λ) there is no solution for any non-trivial initial datum.The Cauchy problem, Ω=ℝN, is also analysed for 1<p<+(λ). We find the same phenomenon about the critical power p+(λ) as above. Moreover, there exists a Fujita-type exponent, F(λ), in the sense that, independently of the initial datum, for 1<p<F(λ), any solution blows up in a finite time. Moreover, F(λ)>1+2/N, which is the Fujita exponent for the heat equation (λ=0).

On backward self-similar blow-up solutions to a supercritical semilinear heat equation

2010 ◽  
Vol 140 (4) ◽  
pp. 821-831 ◽  
Author(s):  
Noriko Mizoguchi
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Blow Up ◽  
Radial Solution ◽  
Similar Solution ◽  
Semilinear Heat Equation ◽  
Blowing Up ◽  
Self Similar Solution ◽  
Self Similar ◽  
Image Position

We are concerned with a Cauchy problem for the semilinear heat equationthen u is called a backward self-similar solution blowing up at t = T. Let pS and pL be the Sobolev and the Lepin exponents, respectively. It was shown by Mizoguchi (J. Funct. Analysis257 (2009), 2911–2937) that k ≡ (p − 1)−1/(p−1) is a unique regular radial solution of (P) if p > pL. We prove that it remains valid for p = pL. We also show the uniqueness of singular radial solution of (P) for p > pS. These imply that the structure of radial backward self-similar blow-up solutions is quite simple for p ≥ pL.

scholarly journals Uniqueness in the Cauchy problem for the heat equation

1999 ◽  
Vol 42 (3) ◽  
pp. 455-468 ◽  
Author(s):  
Soon-Yeong Chung
Keyword(s):  
Cauchy Problem ◽  
Continuous Function ◽  
Heat Equation ◽  
Growth Condition ◽  
Uniqueness Of Solutions ◽  
The Cauchy Problem ◽  
Image Position

We relax the growth condition in time for uniqueness of solutions of the Cauchy problem for the heat equation as follows: Let u(x, t) be a continuous function on ℝn × [0, T] satisfying the heat equation in ℝn × (0, t) and the following:(i) There exist constants a > 0, 0 < α < 1, and C > 0 such that(ii) u(x, 0) = 0 for x ∈ ℝn.Then u(x, t)≡ 0 on ℝn × [0, T]We also prove that the condition 0 < α < 1 is optimal.

Blow-up for quasilinear heat equations with critical Fujita's exponents

1994 ◽  
Vol 124 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Victor A. Galaktionov
Keyword(s):  
Heat Equation ◽  
Critical Exponent ◽  
Critical Case ◽  
Source Term ◽  
Blow Up ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
The Cauchy Problem ◽  
Fixed Constant ◽  
Image Position

We consider the Cauchy problem for the quasilinear heat equationwhere σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivitywith σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.

Non-existence of radial backward self-similar blow-up solutions with sign change

2011 ◽  
Vol 141 (4) ◽  
pp. 825-834
Author(s):  
Noriko Mizoguchi
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Blow Up ◽  
Similar Solution ◽  
Semilinear Heat Equation ◽  
Sign Change ◽  
Blowing Up ◽  
Self Similar Solution ◽  
Self Similar ◽  
Image Position

We consider a Cauchy problem for a semilinear heat equationwith p > 1. If u(x, t) = (T − t)−1/(p−1)ϕ((T − t)−1/2x) for x ∈ ℝN and t ∈ [0, T),where ϕ ∈ L∞(ℝN) is a solution not identically equal to zero ofthen u is called a backward self-similar solution blowing up at t = T. We show that, for all p > 1, there exists no radial sign-changing solution of (E) which belongs to L∞(ℝN). This implies the non-existence of radial backward self-similar solution with sign change blowing up in finite time.

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

On the Cauchy problem associated with the Brinkman flow in

2021 ◽  
pp. 1-20
Author(s):  
Michel Molina Del Sol ◽  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Comparison Principle ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
Brinkman Equations ◽  
The Cauchy Problem ◽  
Model Fluid ◽  
Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

scholarly journals The Cauchy problem for the Finsler heat equation

2020 ◽  
Vol 13 (3) ◽  
pp. 257-278 ◽  
Author(s):  
Goro Akagi ◽  
Kazuhiro Ishige ◽  
Ryuichi Sato
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Heat Equation ◽  
Laplace Operator ◽  
Sufficient Condition ◽  
Smooth Functions ◽  
Dual Norm ◽  
The Cauchy Problem

AbstractLet H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H. Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))}. In this paper we prove that the Finsler–Laplace operator {\Delta_{H}} acts as a linear operator to {H_{0}}-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation\partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0,where {N\geq 1} and {\partial_{t}:=\frac{\partial}{\partial t}}.

scholarly journals Explicit solutions for a system of coupled Lyapunov differential matrix equations

1987 ◽  
Vol 30 (3) ◽  
pp. 427-434 ◽  
Author(s):  
L. Jodar ◽  
M. Mariton
Keyword(s):  
Cauchy Problem ◽  
Explicit Expression ◽  
Linear Quadratic ◽  
The Cauchy Problem ◽  
Matrix Differential Equations ◽  
Lyapunov Matrix ◽  
Error Accumulation ◽  
Matrix Differential ◽  
Image Position ◽  
Differential Matrix

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the typewhere Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

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