<p style='text-indent:20px;'>We discuss some recent results (mostly from the last decade) on stable and finite Morse index solutions of semilinear elliptic equations, where Norman Dancer has made many important contributions. Some open questions in this direction are also discussed.</p>

The decidability of the reachability problem for VASS has been open for a long time, and today the computational complexity of the problem is still open. Obviously, there is more hope of obtaining elementary complexity upper bounds by restricting the class of VASS, for example by considering those for which the reachability sets are effectively semilinear. Presburger arithmetic would then provi...

We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation $ \begin{equation*} u_{t} = \Delta u + |u|^{p} \end{equation*} $ on complete Riemannian manifolds dimension n \geq 5 with nonnegative Ricci tensor, when p is smaller than critical Sobolev exponent \frac{n+2}{n-2} $.

We consider extended variants of spiking neural P systems and show how these extensions of the original model allow for easy proofs of the computational completeness of extended spiking neural P systems and for the characterization of semilinear sets and regular languages by finite extended spiking neural P systems (defined by having only finite sets of rules assigned to the cells) with only a ...

We calculate the universal Euler characteristic and universal dimension function on semilinear and semibounded sets and obtain some criteria for definable equivalence of semilinear and semibounded sets in terms of these invariants.

<p style='text-indent:20px;'>We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of respective undisturbed system. We apply our to a large class nonlinear reaction-diffusion equations comprising Chaffee–Infante equations, example.</p>

In this article, we prove pointwise bounds for solutions to the semilinear wave equation with integer powers $$p\ge 3$$ on Kerr backgrounds small angular momentum and initial data. We expect that proved in paper are optimal.

we consider the semilinear elliptic boundary value problem  = ∈∂ω− δ = ∈ωu x xu x f u x x( ) 0;( ) λ ( ( ));where λ > 0 is a parameter, ω is a bounded region in rn with a smooth boundary, and f is asmooth function. we prove, under some additional conditions, the existence of a positive solution for λlarge. we prove that our solution u for λ large is such that = →∞∈ω|| u ||: sup| u(x) |xas λ ...

The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts I, II. In Part I (this pap...