This paper is devoted to prove analyticity of stable invariant manifold in a neighbourhood of an unstable steady-state solution for GinzburgLandau equation defined in a bounded domain of dimension not more than three. This investigation is made for possible applications in stabilization theory for semilinear parabolic equation. Introduction In this paper we prove analyticity of stable invariant...

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank–Nicolson, and discontinuous Galerkin dG(r) methods are addressed. For their full discretizations, we employ elliptic reconstructions...

This paper is devoted to establishing local and global existence theorems for autonomous semilinear parabolic initial value problems. The local existence theorems do not require Lipchitz condition on nonlinear term. The global existence theorem is an extension of the well-known result of Fujita-Weissler for semilinear heat equations to general autonomous semilinear parabolic equations and systems.

In this article, we study the long time behavior of a phase-field parabolichyperbolic system arising from the phase-field theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation ...

We study the existence and long-time behavior of solutions to a semilinear strongly degenerate parabolic equation on R under an arbitrary polynomial growth order of the nonlinearity. To overcome some significant difficulty caused by the lack of compactness of the embeddings, the existence of global attractors is proved by combining the tail estimates method and the asymptotic a priori estimate ...

We study the null controllability problem for a semilinear parabolic equation, with hysteresis entering in the semilinearity. Under suitable hypotheses, we prove the controllability result and explicitly treat the cases where the hysteresis relationship is given by a Play or a Preisach operator.

In this article, we study the higher-order semilinear parabolic equation ut + (−∆)u = |u|, (t, x) ∈ R+ × R , u(0, x) = u0(x), x ∈ R . Using the test function method, we derive the blow-up critical exponent. And then based on integral inequalities, we estimate the life span of blow-up solutions.

Using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semilinear degenerate parabolic equation involving the Grushin operator in a bounded domain. We assume a polynomial type growth on the nonlinearity, and an exponential growth on the external force. The obtained results extend some existing results for non-autonomous reaction-diffusi...

[2] J. Bourgain, Fourier Transform Restriction Phenomena for Certain Lattice Subsets and Applications to Nonlinear Evolution Equations, Geometric and Functional Analysis 3 (1993), 107–156. [3] M. Christ, J. Colliander and T. Tao, Instability of the periodic nonlinear Schrödinger equation, Preprint, 2003. [4] M. Hochbruck, A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear par...

An optimal control problem for a semilinear parabolic equation is investigated, where pointwise constraints are given on the control and the state. The state constraints are of mixed (bottleneck) type, where associated Lagrange multipliers can assumed to be bounded and measurable functions. Based on this property, a second-order sufficient optimality condition is established that considers stro...