We study the existence and uniqueness of non-negative solutions of the nonlinear parabolic equation

posed in Q = RN × (0, ∞) with general initial data u(x, 0) = u0(x) ≧ 0. We find optimal exponential growth conditions for existence of solutions. Similar conditions apply for uniqueness, but the growth rate is different. Such conditions strongly depart from the linear case m = 1, ut = Δu – u, and also from the purely diffusive case ut = Δum.

Some evidence indicates that spherically symmetric solutions of the compressible Euler equations blow up near the origin at some time under certain circumstances (cf. [4,19]). In this paper, we observe a criterion for L∞ Cauchy data of arbitrarily large amplitude to ensure the existence of L∞ spherically symmetric solutions in the large, which model outgoing blast waves and large-time asymptotic solutions. The equilibrium states of the solutions and their asymptotic decay to such states are analysed. Some remarks on global spherically symmetric solutions are discussed.

The existence of global weak solutions is shown for the equations of isentropic gas dynamics with inhomogeneous terms by the viscosity method. A generalised version of the method of invariant regions is developed to obtain the uniform L∞ bounds of the viscosity solutions, and the method of compensated compactness is applied to show the existence of weak solutions as limits of the viscosity solutions. The lower positive bound for the density function is also obtained. As an example, a hydrodynamic model for semiconductors is analysed

We analyse the dynamics of a prototype model for competing species with diffusion coefficients (d1d2) in a heterogeneous environment Ω. When diffusion is switched off, at each point x ∊ Ω we have a pair of ODE's: the kinetic. If for some x ∊ Ω kinetic has a unique stable coexistence state, we show that there exist such that for every the RD-model is persistent, in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there exist xu, xv ∊ Ω such that the semitrivial coexistence states (u, 0) and (0, v) of the kinetic are globally asymptotically stable at x = xu and x = xv, respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of xspopulations can be found for some range of the diffusion coefficients.

Singular perturbation techniques, monotone schemes, fixed point index, global analysis of persistence curves, global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems.

We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model has three or more coexistence states.

In this paper, we give some new examples of the energy gap phenomenon for functionals defined in Sobolev spaces. Our result is independent of that of Giaquinta, Modica and Soucek. We also give some new characterisations of Sobolev maps which can be approximated by smooth maps.

Suppose that f is meromorphic of finite order in the plane, and that f″ has only finitely many zeros. We prove a strong estimate for the frequency of distinct poles of f. In particular, if the poles of f have bounded multiplicities, then f has only finitely many poles.

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem for

is locally well posed in the largest space where the three conservations

can be justified. Here E(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

We consider the nonlinear elliptic problem ± (Δu + λu) + f(x, u) = h(x) in Ω, u = 0 on ∂Ω, where Ω is a bounded smooth domain in ℝN, λ is near the first eigenvalue and h(x) is orthogonal to the first eigenfunction. We give some conditions of existence of positive solutions and of multiple solutions in terms of the primitive of f with respect to u.

The purpose of this paper is to study boundary value problems for elliptic pseudodifferential operators which originate from the problem of existence of Markov processes in probability theory, generalising some results of our previous work. Our approach has a great advantage of intuitive interpretation of sufficient conditions for the unique solvability of boundary value problems in terms of Markovian motion. In fact, we prove that if a Markovian particle moves incessantly both by jumps and continuously in the state space, not being trapped in the set where no reflection phenomenon occurs, then our boundary value problem is uniquely solvable in the framework of Sobolev spaces of LP style.

The cohomology of the classifying space of the gauge group of the nontrivial SO(3) bundle over S2 has no torsion.

We construct nontrivial, non-negative quasiconvex functions denned on M2×2 with p-th order growth such that the zero sets of the functions are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. We assume that the graphs do not have rank-one connections with the Lipschitz constants sufficiently small. In particular, we are able to construct quasiconvex functions which are homogeneous of degree p (p > 1) and ‘conjugating’ invariant.

In this paper we use stochastic semiclassical analysis and the logarithmic transformation to study the gradients of the approximate travelling wave solutions for the generalised KPP equations with Gaussian and Dirac delta initial distributions. We apply the logarithmic transformation to the nonlinear reaction diffusion equations and obtain a Maruyama–Girsanov–Cameron–Martin formula for the drift μ2 ∇ log uμ, uμ being a solution of a generalised KPP equation. We obtain that μ2|∇ log uμ(t,x)| is bounded and the trough is flat. The difficult problem in this paper is to prove that the corresponding crest is flat. A probabilistic approach is used in this paper to treat this problem successfully.