Peter Poláčik

Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems

In this paper, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent. ∗Supported in part by NSF Grant DMS-0400702 †Supported in part by VEGA Grant 1/3021/06

Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations

A vector bundle morphism of a vector bundle with strongly ordered Banach spaces as fibers is studied. It is assumed that the fiber maps of this morphism are compact and strongly positive. The existence of two complementary, dimension-one and codimension-one, continuous subbundles invariant under the morphism is established. Each fiber of the first bundle is spanned by a positive vector (that is, a nonzero vector lying in the order cone), while the fibers of the other bundle do not contain a positive vector. Moreover, the ratio between the norms of the components (given by the splitting of the bundle) of iterated images of any vector in the bundle approaches zero exponentially (if the positive component is in the denominator). This is an extension of the Krein-Rutman theorem which deals with one compact strongly positive map only. The existence of invariant bundles with the above properties appears to be very useful in the investigation of asymptotic behavior of trajectories of strongly monotone discrete-time dynamical systems, as demonstrated by Poláčik and Tereščák (Arch. Ration. Math. Anal.116, 339–360, 1991) and Hess and Poláčik (preprint). The present paper also contains some new results on typical asymptotic behavior in scalar periodic parabolic equations.

Complicated dynamics of scalar reaction diffusion equations with a nonlocal term

We consider the dynamics of scalar equations u t , = u xx + f ( x , u ) + c ( x )α( u ), 0 x f , c , α appropriately, it is shown that complicated dynamics can occur. Specifically, linearisations at equilibria can have any number of purely imaginary eigenvalues. Moreover, the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily, up to any finite order. These results are in marked contrast with the case α = 0, where bounded solutions are known to converge to equilibrium.

Competing Species near a Degenerate Limit

We consider a competitive reaction-diffusion model of two species in a bounded domain which are identical in all aspects except for their birth rates, which differ by a function g. Under a fairly weak hypothesis, the semitrivial solutions always exist. Our analysis provides a description of the stability of these solutions as a function of the diffusion rate

CONVERGENCE TO EQUILIBRIUM FOR SEMILINEAR PARABOLIC PROBLEMS IN

\mu

Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains

and the difference between the birth rates g. In the case in which the magnitude of g is small we provide a fairly complete characterization of the stability in terms of the zeros of a single function. In particular, we are able to show that for any fixed number n, one can choose the difference function g from an open set of possibilities in such a way that the stability of the semitrivial solutions changes at least n times as the diffusion

Boundedness of prime periods of stable cycles and convergence to fixed points in discrete monotone dynamical systems

\mu

IMMEDIATE REGULARIZATION AFTER BLOW-UP

is varied over

Monotonicity of stable solutions in shadow systems

(0,\infty)

Symmetry and convergence properties for non-negative solutions of nonautonomous reaction–diffusion problems

. This result allows us to make conclusions concerning the existence of coexistence states. Furthermore, we show that under these hypotheses, coexistence states are unique if ...