Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity

We study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, we obtain the existence of global solution and blow-up at +∞ under some suitable conditions. On the other hand, the results for decay estimates of the global solutions are also given. Our result in this paper means that the polynomial nonlinearity is a critical condition of blow-up in finite time for the solutions of semilinear heat equations.     

Lower bounds for blow-up time in parabolic problems under Dirichlet conditions

We consider an initial-boundary value problem for the semilinear heat equation whose solution may blow up in finite time. We use a differential inequality technique to determine a lower bound on blow-up time if blow-up occurs. A second method based on a comparison principle is also presented.     

Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions

A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in n>1 dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.     

Well-posedness of the fractional Ginzburg–Landau equation

In this paper, we investigate the well-posedness of the real fractional Ginzburg–Landau equation in several different function spaces, which have been used to deal with the Burgers’ equation, the semilinear heat equation, the Navier–Stokes equations, etc. The long time asymptotic behavior of the nonnegative global solutions is also studied in details.     

Parallel Schwarz waveform relaxation method for a semilinear heat equation in a cylindrical domain

We present here a proof of well-posedness and convergence for the parallel Schwarz waveform relaxation algorithm adapted to the semilinear heat equation in a cylindrical domain. It relies on a careful estimate of a local time of existence thanks to the Banach theorem in a well chosen metric space, together with new cylindrical error estimates.     

Lower bounds for blow-up time in parabolic problems under Neumann conditions

We consider an initial boundary value problem for the semilinear heat equation under homogeneous Neumann boundary conditions in which the solution may blow up in finite time. A lower bound for the blow-up time is determined by means of a differential inequality argument when blow up occurs. Under alternative conditions on the nonlinearity, some additional bounds for blow-up time are also determined.     

Pullback attractors for a semilinear heat equation in a non-cylindrical domain

The existence and uniqueness of a variational solution satisfying energy equality is proved for a semilinear heat equation in a non-cylindrical domain with homogeneous Dirichlet boundary condition, under the assumption that the spatial domains are bounded and increase with time. In addition, the non-autonomous dynamical system generated by this class of solutions is shown to have a global pullback attractor.     

Blow-up in parabolic problems under Robin boundary conditions

By means of a first-order differential inequality technique, sufficient conditions are determined which imply that blow-up of the solution does occur or does not occur for the semilinear heat equation under Robin boundary conditions. In addition, a lower bound on blow-up time is obtained when blow-up does occur.     

Life span of positive solutions for a semilinear heat equation with general non-decaying initial data

We prove upper bounds on the life span of positive solutions for a semilinear heat equation. For non-decaying initial data, it is well known that the solutions blow up in finite time. We give two types estimates of the life span in terms of the limiting values of the initial data in space.     

A note on approximate controllability for semilinear one-dimensional heat equation

The one-dimensional semilinear heat equation is considered. It is shown that if the nonlinear functionF(y) is uniformly bounded then the system is approximately controllable for every given terminal timeT>0 under some ordinary condition onb. The results may be extended to the general one-dimensional semilinear heat equation with one-dimensional control or to a boundary control heat system with semilinear boundary condition.     

Remarks on a semilinear heat equation with integral boundary conditions

For a semilinear heat equation we consider a nonlocal boundary problem. On the basis of the solution of a Dirichlet problem for a parabolic equation and Volterra integral equation we establish the well-posedness for the nonlocal problem, which generalizes some recent results.     

由退化抛物方程支配串联系统的零能控性

该文讨论了由一个半线性退化抛物方程与半线性热方程构成的串联系统的零能控性. 这里控制函数仅施加在一个方程上. 证明的关键是建立适当的能观性不等式.     

Various behaviors of solutions for a semilinear heat equation after blowup

The present paper is concerned with a Cauchy problem for a semilinear heat equation

(P)     

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.     

Boundary characteristic point regularity for semilinear reaction-diffusion equations: towards an ode criterion

The classical problem of regularity of boundary characteristic points for semilinear heat equations with homogeneous Dirichlet conditions is considered. The Petrovskii ( 2?{loglog} ) \left( {2\sqrt {{\log \log }} } \right) criterion (1934) of the boundary regularity for the heat equation can be adapted to classes of semilinear parabolic equations of reaction–diffusion type and takes the form of an ordinary differential equation (ODE) regularity criterion. Namely, after a special matching with a boundary layer, the regularity problem reduces to a onedimensional perturbed nonlinear dynamical system for the first Fourier-like coefficient of the solution in an inner region. A similar ODE criterion, with an analogous matching procedures, is shown formally to exist for semilinear fourth order biharmonic equations of reaction-diffusion type. Extensions to regularity problems of backward paraboloid vertices in \mathbbR^N {\mathbb{R}^N} are discussed. Bibliography: 54 titles. Illustrations: 1 figure.     

移动边界半线性热传导方程始边值问题分析近似解

本文应用Fourier方法求得移动边界非齐次线性热传导方程始边值问题解及半线性方程问题分析近似解     

Quenching Versus Blow-up

This paper is concerned with the semilinear heat equation u_t = Δu - u^{-q} in Ω × (0, T) under the nonlinear boundary condition \frac{∂u}{∂v} = u^p on ∂Ω × (0, T). Criteria for finite time quenching and blow-up are established, quenching and blow-up sets are discussed, and the rates of quenching and blow-up are obtained.     

Weak approximation of the complex Brownian sheet from a Lévy sheet and applications to SPDEs

We consider a Lévy process in the plane and we use it to construct a family of complex-valued random fields that we show to converge in law, in the space of continuous functions, to a complex Brownian sheet. We apply this result to obtain weak approximations of the random field solution to a semilinear one-dimensional stochastic heat equation driven by the space–time white noise.     

Asymptotic behavior of a semilinear problem in heat conduction with memory

This paper is devoted to existence, uniqueness and asymptotic behavior, as time tends to infinity, of the solutions of an integro-partial differential equation arising from the theory of heat conduction with memory, in presence of a temperature-dependent heat supply. A linearized heat flux law involving positive instantaneous conductivity is matched with the energy balance, to generate an autonomous semilinear system subject to initial history and Dirichlet boundary conditions. Existence and uniqueness of solution is provided. Moreover, under proper assumptions on the heat flux memory kernel, the existence of absorbing sets in suitable function spaces is achieved. Received March 23, 1997 - Revised version received November 12, 1997     

半线性热方程的反馈零能控性

考虑具有Lipschitz非线性项,半线性热方程的最优控制问题.我们将运用观测不等式,证明值函数ψ作为相应Hamilton-Jacobi方程的唯一粘性正解是局部Lipschitz连续的.最后,运用动态规划方法,得到系统最优的反馈控制.