Semilinear Parabolic Initial-boundary Value Research Articles

Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder

Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.

Wavelet based quasilinearization method for semi-linear parabolic initial boundary value problems

In this paper, numerical methods based on quasilinearization and Haar and Legendre wavelets to solve a class of semi linear parabolic initial boundary value problem (SPIBVP) have been presented. The Haar and Legendre wavelet methods have been successfully combined with quasilinearization to solve SPIBVP efficiently. The presented numerical scheme has been illustrated using appropriate examples including Fisher equation and the obtained results show that the proposed numerical scheme is robust and easy to apply.

Local and global solutions for some parabolic nonlocal problems

We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a “Newtonian” nonlocal term.

Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities

In this paper we discuss continuation properties and asymptotic behavior of ɛ-regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions. A necessary and sufficient condition for global in time existence of ɛ-regular solutions is given. We also formulate sufficient conditions to construct a piecewise ɛ-regular solutions (continuation beyond maximal time of existence for ɛ-regular solutions). Applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed. In particular global solvability and the existence of a global attractor for u t t + η ( − Δ D ) 1 2 u t + ( − Δ D ) u = f ( u ) in H 0 1 ( Ω ) × L 2 ( Ω ) is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition. Similar result is obtained for a 2 mth order semilinear parabolic initial boundary value problem in a Hilbert space H 2 , { B j } m ( Ω ) .

Recovery of An Unknown Flux in Parabolic Problems with Nonstandard Boundary Conditions: Error Estimates

In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain Ω ⊂ ℝN, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant α(t), accompanied with a nonlocal (integral) Dirichlet side condition.We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution u and also of the unknown function α.

Determination of a Robin coefficient in semilinear parabolic problems by means ofboundary measurements

We consider a semilinear parabolic initial boundary value problemof second order in a bounded domain Ω ⊂ ℝNwith a Lipschitz continuous boundary. We study the problem ofdetermination of the Robin coefficient at the boundary part Γnonfrom a nonlocal boundary condition. We prove the well posedness of the problem.We design a numerical scheme for its approximation and perform the erroranalysis.

Galerkin/Runge–Kutta Discretizations for Semilinear Parabolic Equations

A new class of fully discrete Galerkin/Runge–Kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems. Unlike any classical counterpart, this class offers arbitrarily high-order convergence without suffering from what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Furthermore, it is noted that special Runge–Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low-order method.

Blow-up of solutions of semilinear parabolic differential equations

For a semilinear parabolic initial boundary value problem we establish criterions on blow-up of the solution in finite time and give bounds for the blow-up time. We treat several applications in both finite and infinite domains. For comparison, sufficient conditions are also given for the existence of global solutions.

Monotone method for second order periodic boundary value problems

IN RECENT years, there has been an extensive study of the existence of periodic solutions [l, S-11, 14, 151. In [8, 111, the existence of solutions of first and second order PBVP (period boundary value problems) has been obtained by a novel approach of combining the classical method of lower and upper solutions and the method of alternative problems (LyapunovSchmidt method), which provide conditions that are easily verifiable and which covers previous known results of other authors. As a constructive method for obtaining extremal solutions of initial and boundary value problems, the monotone iterative procedure has been employed by several researchers [5-7, 11-13, 151. The objective of this paper is to employ this useful technique for second order PBVP to obtain the minimal and maximal solutions as limits of monotone iterates. Our method can be used to study semilinear parabolic initial boundary value problems and other problems at resonance.

Asymptotic Behavior of Solutions of a Class of Nonlinear Differential Equations in Banach Space

A semi-linear ordinary differential equation in a Banach space is considered. The coefficient operator $A(t)$ has the domain $D(A)$ which is independent of t and not necessarily dense. It is shown that the evolution operator $U(t,s)$ corresponding to the unperturbed linear equation has an integrable singularity at $t = s$ and is strongly continuously differentiable in s on $D(A)$. Such examples have been obtained by W. von Wahl [13] and H. Kielhöfer [4], [5], and are also obtained in this paper. The nonlinear term satisfies either a uniform or local Lipschitz’s condition with respect to the unknown solution. The principal tools are the semigroup theory and integral inequalities. Several results on the asymptotic behavior of the solution of the semi-linear equation are obtained. These results are applicable to the problem of the stability of semi-linear parabolic initial boundary value problems within the framework of the $C^\alpha $-theory.