scholarly journals The boundary behavior and uniqueness of solutions of the heat equation.

1960 ◽  
Vol 94 (3) ◽  
pp. 337-337
Author(s):  
F. W. Gehring
Keyword(s):  
Heat Equation ◽  
Boundary Behavior ◽  
Uniqueness Of Solutions

scholarly journals The Yang-Mills heat equation with finite action in three dimensions

2022 ◽  
Vol 275 (1349) ◽  
Author(s):  
Leonard Gross
Keyword(s):  
Heat Equation ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Three Dimensions ◽  
Dimensional Manifold ◽  
Uniqueness Of Solutions ◽  
Reconstruction Procedure ◽  
Content Type ◽  
Yang Mills ◽  
Gauge Functions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

scholarly journals A local Hopf lemma for solutions of the one dimensional heat equation

1997 ◽  
Vol 146 ◽  
pp. 1-12
Author(s):  
Noriaki Suzuki
Keyword(s):  
Heat Equation ◽  
Finite Order ◽  
Boundary Point ◽  
Local Minimum ◽  
Boundary Behavior ◽  
Lateral Boundary ◽  
Temperature Function ◽  
One Dimensional ◽  
The One

Abstract.The boundary behavior of solutions of the heat equation (temperature functions) is investigated. It is proved that a temperature function is identically equal to zero if it vanishes of finite order at some lateral boundary point where it attains a local minimum.

Existence and uniqueness of solutions to a concentrated capacity problem with change of phase

1990 ◽  
Vol 1 (4) ◽  
pp. 339-351 ◽  
Author(s):  
Daniele Andreucci
Keyword(s):  
Heat Equation ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Weak Sense ◽  
Uniqueness Of Solutions ◽  
Limiting Case ◽  
Multidimensional Domain ◽  
Diffusion Problems

A concentrated capacity problem is posed for the heat equation in a multidimensional domain. In the concentrated capacity (i.e. in a portion of the boundary of the domain) a change of phase takes place, and a Stefan-like problem is posed. This scheme has been introduced in the literature as the formal limiting case of a certain class of diffusion problems.Our main result is a theorem of continuous dependence of the solution on the data. It is also used to prove the existence of the solution (in a weak sense), assuming only integrability of the data. The solution is found as the limit of the solutions of the approximating problems.

Uniqueness and non-uniqueness of solutions of the boundary value problems of the heat equation

2015 ◽  
Author(s):  
Meiramkul M. Amangaliyeva ◽  
Muvasharkhan T. Jenaliyev ◽  
Minzilya T. Kosmakova ◽  
Murat I. Ramazanov
Keyword(s):  
Heat Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Uniqueness Of Solutions

The Cauchy problem for a system of nonlinear heat equations in two space dimensions

2015 ◽  
Vol 08 (01) ◽  
pp. 1550004
Author(s):  
Amel Chouichi ◽  
Sarah Otsmane
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Dimensional Space ◽  
Local Existence ◽  
Energy Functional ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Uniqueness Of Solutions ◽  
Semilinear Heat Equation ◽  
Singular Initial Data

This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity in two-dimensional space which is the analogue of the scalar model problem studied in [S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Local well posedness of a 2D semilinear heat equation, Bull. Belg. Math. Soc.21 (2014) 1–17]. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space (H1× H1)(ℝ2). The uniqueness part is nontrivial although it follows Brezis–Cazenave's proof [H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math.68 (1996) 73–90] in the case of monomial nonlinearity in dimension d ≥ 3. Next, we show that in the defocusing case our solution is bounded, and therefore exists globally in time. Finally, for this system, we treat the question of blow-up in finite time under the negativity condition on the energy functional. The technique to be used is adapted from [Bull. Belg. Math. Soc. 21 (2014) 1–17].

scholarly journals On the evolution equation with a dynamic Hardy-type potential

2021 ◽  
Author(s):  
Jann-Long Chern ◽  
Gyeongha Hwang ◽  
Jin Takahashi ◽  
Eiji Yanagida
Keyword(s):  
Integral Equation ◽  
Singular Point ◽  
Heat Equation ◽  
Existence And Uniqueness ◽  
Singular Potential ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Hardy Type ◽  
Radial Heat ◽  
Type Potential

Abstract Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, non-existence and uniqueness of solutions, and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

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.

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