In this paper, we consider a semilinear parabolic equation ut = ∆u + u q ∫ t 0 u(x, s)ds, x ∈ Ω, t > 0 with nonlocal nonlinear boundary condition u|∂Ω×(0,+∞) = ∫ Ω φ(x, y)u (y, t)dy and nonnegative initial data, where p, q ≥ 0 and l > 0. The blow-up criteria and the blow-up rate are obtained.

A generalized Cauchy-Kowalevski approach is proposed for flatness-based trajectory planning for boundary controlled semilinear systems of PDEs in a one-dimensional spatial domain. For this, the ansatz presented in [16] using formal integration is generalized towards a unified design framework, which covers linear and semilinear PDEs including rather broad classes of nonlinearities arising in ap...

A generalized Cauchy-Kowalevski approach is proposed for flatness-based trajectory planning for boundary controlled semilinear systems of PDEs in a one-dimensional spatial domain. For this, the ansatz presented in [16] using formal integration is generalized towards a unified design framework, which covers linear and semilinear PDEs including rather broad classes of nonlinearities arising in ap...

An analysis of discretizations of the Helmholtz equation in L 2 and in negative norms (extended version) Flatness of semilinear parabolic PDEs-A generalized Chauchy-Kowalevski approach 27/2012 R. Donninger and B. Schörkhuber Stable blow up dynamics for energy supercritical wave equations 26/2012 P.

We study semilinear evolution equations [Formula: see text] posed on a Hilbert space [Formula: see text], where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from [Formula: see text] to itself, and [Formula: see text], [Formula: see text], [Formula: see text]. In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation wi...

We compute the Morse index m (up) of any radial solution up semilinear problem: (P)−Δu=|x|α|u|p−1uinBu=0on∂Bwhere B is unit ball R2 centered at origin, α≥0 fixed and p>1 sufficiently large. In case α=0, i.e. for Lane–Emden problem, this leads to following formula m(up)=4m2−m−2,for p large enough, where number nodal domains u.

We investigate finite deterministic automata in sets with non-homogeneous atoms: integers with successor. As there are uncount-ably many deterministic finite automata in this setting, we restrict our attention to automata with semilinear transition function. The main results is a minimization procedure for semilinear automata. The proof is subtle and refers to decidability of existential Presbu...

The existence of a unique strong solution for the Cauchy problem to semilinear nondegenerate fractional differential equation and for the generalized Showalter–Sidorov problem to semilinear fractional differential equation with degenerate operator at the Caputo derivative in Banach spaces is proved. These results are used for search of solution existence conditions for a class of optimal contro...

In this paper, we consider the question of controllability of a class of linear and semilinear evolution equations on Hilbert space with measures as controls. We present necessary and sufficient conditions for weak and exact (strong) controllability of a linear system. Using this result we prove that exact controllability of the linear system implies exact controllability of a perturbed semilin...

The paper gives a different proof for exitence of ground state solutions to the semilinear problem −∆u + u = b(x)up, p > 2, on RN , where b is invariant with respect to a free acting subgroup of O(N) with m elements. While the standard concentration compactness argument suggests the solvability condition inf b(x)/b∞ ≥ 1, symmetric ground state solutions exist whenever inf b(x)/b∞ > δm, where δm...