On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent

This paper is concerned with a semilinear parabolic equation involving critical Sobolev exponent in a ball or in RN. The asymptotic behavior of unbounded, radially symmetric, nonnegative global solutions which do not decay to zero is given. The structure of the space of initial data is also discussed.     

Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity

We consider the blowup rate of solutions for a semilinear heat equation

     

Large time behaviors of hot spots for the heat equation with a potential

We consider the Cauchy problem of the heat equation with a potential which behaves like the inverse square at infinity. In this paper we study the large time behavior of hot spots of the solutions for the Cauchy problem, by using the asymptotic behavior of the potential at the space infinity.     

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

Life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity

We present a new upper bound of the life span of positive solutions of a semilinear heat equation for initial data having positive limit inferior at space infinity. The upper bound is expressed by the data in limit inferior, not in every direction, but around a specific direction. It is also shown that the minimal time blow-up occurs when initial data attains its maximum at space infinity.     

A singular solution with smooth initial data for a semilinear parabolic equation

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. Our concern in this paper is the existence of a singular solution with smooth initial data. By using the Haraux-Weissler equation, it is shown that there exist singular forward self-similar solutions. Using this result, we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data.     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation but much less on those of the Hermite heat equation due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005). Supported partially by 973 project (2004CB318000)     

Exponential convergence for a periodically driven semilinear heat equation

We consider a semilinear heat equation in one space dimension, with a periodic source at the origin. We study the solution, which describes the equilibrium of this system and we prove that, as the space variable tends to infinity, the solution becomes, exponentially fast, asymptotic to a steady state. The key to the proof of this result is a Harnack type inequality, which we obtain using probabilistic ideas.     

Rates of convergence to zero for a semilinear parabolic equation with a critical exponent

We study nonnegative solutions to the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the sense of Joseph and Lundgren. We establish the rate of convergence to zero of solutions that start from initial data which are near the singular steady state. In the critical case, this rate contains a logarithmic term which does not appear in the supercritical case and makes the calculations more delicate.     

The Fujita exponent for semilinear heat equations with quadratically decaying potential or in an exterior domain

Consider the equation

(0.1)     

On a stochastic hyperbolic integro-differential equation

In this paper we study an initial-boundary-value problem for a hyperbolic integro-differential equation with random memory and a random noise. We establish the existence, uniqueness and exponential stability of solutions. Our method consists of finite-dimensional approximation and energy estimates.     

Sharp estimates of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity

We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with supercritical nonlinearity. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its sharp convergence rate for a class of initial data. We also derive a universal lower bound of the convergence rate which implies the optimality of the result. Proofs are given by a comparison method based on matched asymptotics expansion.     

Blow-up phenomena for a doubly degenerate equation with positive initial energy

In this paper, an initial boundary value problem related to the equation

     

A Pizzetti-type formula for the heat operator

We provide an asymptotic expansion of the integral mean of a smooth function over the Heat ball. Namely we generalize to the Heat operator the so-called Pizzetti’s Formula, which expresses the integral mean of a smooth function over an Euclidean ball in terms of a power series with respect to the radius of the ball having the iterated of the ordinary Laplace operator as coefficients. Similarly here, we express the heat integral mean as a power series with respect to the radius of the heat ball, whose coefficients are powers of a distorted heat operator. We also discuss sufficient conditions to have a finite sum. Received: 27 May 2005     

Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations

We first study the initial value problem for a general semilinear heat equation. We prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low (n?10) or if the steady state is flat enough at infinity: the solution of the heat equation either becomes unbounded as t approaches the lifespan, or eventually stays above or below another bounded radial steady state, depending on if the initial value is above or below the first steady state; moreover, the second steady state must be a constant if n?10.Using this instability result, we then prove that every nonconstant radial steady state of the generalized Fisher equation is a hair-trigger for two kinds of dynamical behavior: extinction and spreading. We also prove more criteria on initial values for these types of behavior. Similar results for a reaction-diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained.     

On symmetries of the heat equation

Two aims are pursued in this article. The first one is methodological and consists of demonstrating the symmetry calculation techniques based on the commutation relation on the example of the quasi-linear heat equation. The second one consists of investigating the following open question: is the algebra of higher symmetries of the linear heat equation exhausted by those which may be obtained by means of the classical symmetries and well-known recursion operators? A positive solution of this question is given.