Unspecified Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-22758 Originally published at: Chipot, M; Fila, M; Quittner, P (1991). Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Mathematica Universitatis Comenianae. New Series, 60(1):35-1...

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation ut = ∆u + |u|p−1u. We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence ...

We prove local existence and uniqueness of the solution (u, ut) ∈ C 0([0, T ];H1 × L2(RN )) of the semilinear wave equation utt −∆u = ut|ut| p−1 for 1 < p < 1 + 2 N .

The slope filtration theorem gives a partial analogue of the eigenspace decomposition of a linear transformation, for a Frobenius-semilinear endomorphism of a finite free module over the Robba ring (the ring of germs of rigid analytic functions on an unspecified open annulus of outer radius 1) over a discretely valued field. In this paper, we give a third-generation proof of this theorem, which...

This paper concerns the study of the numerical approximation for the following initial-boundary value problem: (P)  ut = uxx − a|u|p−1u, 0 < x < 1, t > 0, ux(0, t) = 0 ux(1, t) + b|u(1, t)|q−1u(1, t) = 0, t > 0, u(x, 0) = u0(x) > 0, 0 ≤ x ≤ 1, where a > 0, b > 0 and q > p > 1. We show that the solution of a semidiscrete form of (P ) goes to zero as t goes to infinity and give its asymptotic...

where Ω ⊂ R (N ≥ 1) is a bounded smooth domain, λk is an eigenvalue of the problem −Δu = λu in Ω, u = 0 on ∂Ω, and f : Ω × R → R is a Carathéodory function. If f(x, 0) = 0 for a.e. x ∈ Ω the constant u = 0 is a trivial solution of the problem (P). In this case, the key point is proving the existence of nontrivial solutions for (P). For this purpose, we need to introduce some conditions on the b...

In the standard definition of a P system, a computation step consists of a parallel application of a “maximal” set of nondeterministically chosen rules. Referring to this system as a parallel P system, we consider in this paper a sequential P system, in which each step consists of an application of a single nondeterministically chosen rule. We show the following: 1. For 1-membrane catalytic sys...

The Orbit Problem consists of determining, given a linear transformation A on Q, together with vectors x and y, whether the orbit of x under repeated applications of A can ever reach y. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s. In this paper, we are concerned with the problem of synthesising suitable invariants P ⊆ R, i.e., sets that are stable under A a...

Efficient numerical algorithms are proposed for a class of forward-backward stochastic differential equations (FBSDEs) connected with semilinear parabolic partial differential equations. As in [J. Douglas, Jr., J. Ma, and P. Protter, Ann. Appl. Probab., 6 (1996), pp. 940–968], the algorithms are based on the known four-step scheme for solving FBSDEs. The corresponding semilinear parabolic equat...