Approximate symmetry of a nonlinear heat equation

Using the idea of asymptotic methods of Krylov-Bogolyubov-Mitropol'skii, we study the approximate Galilean symmetry of a multidimensional nonlinear heat equation.Translated from Ukrainsklii Matematicheskii Zhurnal, Vol 43, No. 6, pp. 833–837, September 1991.     

Numerical blow-up for a nonlinear heat equation

This paper concerns the study of the numerical approximation for the following initialboundary value problem

Elliptic regularizations for the nonlinear heat equation

The purpose of this paper is to study two elliptic regularizations for the nonlinear heat equation with nonlinear boundary conditions formulated below. Asymptotic expansions of the order zero for the solutions of these elliptic regularizations are established, including some boundary layer corrections. Under some appropriate smoothness and compatibility conditions on the data estimates for the remainder terms with respect to the C([0,T];L²(Ω)) norm are proved in order to validate these expansions.     

On a nonlinear heat equation with functional dependence

We reduce the Cauchy problem for a heat equation with the nonlinear right-hand side which depends on some functionals to an equivalent integral equation. Considering mainly Banach spaces of continuous, bounded and exponentially bounded functions, we give some natural sufficient conditions for the existence and uniqueness of solutions to these equations. We give a counterexample which shows that the Lipschitz condition is, in general, insufficient for the Cauchy problem with unbounded data and with functional dependence to guarantee an existence result     

On some method for solving a nonlinear heat equation

Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation.     

Loss of positivity in a nonlinear scalar heat equation

Using the theory of fixed point index, we discuss the existence of nontrivial (multiple) solutions of a nonlinear scalar heat equation with nonlocal boundary conditions depending on a positive parameter. Solutions lose positivity as the parameter decreases. For a certain parameter range, not all solutions can be positive but there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.     

A one-dimensional nonlinear heat equation with a singular term

In this paper we are concerned with the Dirichlet problem for the one-dimensional nonlinear heat equation with a singular term:

     

Universal bound for global solution of nonlinear heat equation

Positivity - The aim of this paper is to study some properties of positive solutions to the nonlinear diffusion equation $$\begin{aligned} \frac{\partial u(x,t)}{\partial t} = \Delta _p u(x,t) +...     

The heat equation with nonlinear generalized Robin boundary conditions

Let Ω⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H¹-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))_t?0 on L²(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v∈Lp(Ω), p?2 and every t>0, one has

     

Subcritical Kuramoto-Sivashinsky-type equation on a half-line

In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto-Sivashinsky-type equation

(0.1)     

Sharp estimates for approximations to a nonlinear backward heat equation

A nonlinear backward heat problem for an infinite strip is considered. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.     

Global actions of Lie symmetries for the nonlinear heat equation

By restricting to a natural class of functions, we show that the Lie point symmetries of the nonlinear heat equation exponentiate to a global action of the corresponding Lie group. Remarkably, in most of the cases, the action turns out to be linear.     

Dynamics of the nonlinear stochastic heat equation with singular perturbation

The approximation in probability for a singular perturbed nonlinear stochastic heat equation is studied. First the approximation result in the sense of probability is obtained for solutions defined on any finite time interval. Furthermore it is proved that the long time behavior of the stochastic system is described by a global random attractor which is upper semi-continuous with respect to the singular perturbed parameter. This also means the long time effectivity of the approximation with probability one.