Characteristic Planes and Pointwise Solutions of the Heat Equation

The Pointwise Estimates of Solutions for Semilinear Dissipative Wave Equation

In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the Cauchy problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the pointwise estimates of the solution. keywords: semilinear dissipative wave equation, pointwise estimates, Green function. MSC...

Unbounded solutions of the nonlocal heat equation

We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: ut = J ∗u−u , where J is a symmetric continuous probability density. Depending on the tail of J , we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a su...

Stability and asymptotic behaviour of solutions of the heat equation

where Ω is a bounded and smooth subset of Rn , n 1, m > 0 and p 1. Problem (1.1)–(1.3) (see Galaktionov, 1981; Samarskii et al., 1995) describes the propagation of thermal perturbations in a medium with a nonlinear heat conduction coefficient and a heat source depending on the temperature when u0 0. Local existence for the solutions of (1.1)–(1.3) has been proved when m > 1 (the so-called slow ...

Diagonal and Monomial Solutions of the Matrix Equation AXB=C

In this article, we consider the matrix equation $AXB=C$, where A, B, C are given matrices and give new necessary and sufficient conditions for the existence of the diagonal solutions and monomial solutions to this equation. We also present a general form of such solutions. Moreover, we consider the least squares problem $min_X |C-AXB |_F$ where $X$ is a diagonal or monomial matrix. The explici...

WAVELET SOLUTIONS OF THE KLEIN-GORDON EQUATION