Semilinear elliptic equations with singular nonlinearities

We prove existence, regularity and nonexistence results for problems whose model is $$-\Delta u = \frac{f(x)}{u^{\gamma}}\quad {{\rm in}\,\Omega},$$ with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of ${\mathbb{R}^{N}}$ . Here γ > 0 and f is a nonnegative function on Ω. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of γ (which can be equal, larger or smaller than 1).     

Method for solving hyperbolic systems with initial data on non-characteristic manifolds with applications to the shallow water wave equations

We are concerned with hyperbolic systems of order-one linear PDEs originated on non-characteristic manifolds. We put forward a simple but effective method of transforming such initial conditions to standard initial conditions (i.e. when the solution is specified at an initial moment of time). We then show how our method applies in fluid mechanics. More specifically, we present a complete solution to the problem of long waves run-up in inclined bays of arbitrary shape with nonzero initial velocity.     

Semilinear equations with exponential nonlinearity and measure data

We study the existence and non-existence of solutions of the problem

(0.1)

where Ω is a bounded domain in , N3, and μ is a Radon measure. We prove that if , then (0.1) has a unique solution. We also show that the constant 4π in this condition cannot be improved.     

Semilinear hyperbolic systems violating the null condition

We consider systems of semilinear wave equations in three space dimensions with quadratic nonlinear terms not satisfying the null condition. We prove small data global existence of the classical solution under a new structural condition related to the weak null condition. For two-component systems satisfying this condition, we also observe a new kind of asymptotic behavior: Only one component is dissipated and the other one behaves like a non-trivial free solution in the large time.     

Semilinear elliptic equations and systems with diffuse measures

We study the equation −, where g(·, s) is finite outside sets of zero H ¹-capacity, , and μ is a diffuse measure. As an application, we provide a positive answer to a question of Lucio Boccardo concerning existence of solutions of an elliptic system with absorption.

A Lucio “Tu se’ lo mio maestro e ’l mio autore” (Dante, Inferno, I, 85)

     

Semilinear elliptic systems with measure data

Annali di Matematica Pura ed Applicata (1923 -) - We study the Dirichlet problem for systems of the form $$-\varDelta u^k=f^k(x,u)+\mu ^k,\,x\in \varOmega ,\,k=1,\ldots ,n$$ , where $$\varOmega...     

The heat equation with singular nonlinearity and singular initial data

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u₀ is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u₀?γdΩ for some γ>0. We find a unique solution in the space .     

Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation

We investigate totally linearly degenerate hyperbolic systems with relaxation. We aim to study their semilinear behavior, which means that the local smooth solutions cannot develop shocks, and the global existence is controlled by the supremum bound of the solution. In this paper we study two specific examples: the Suliciu-type and the Kerr-Debye-type models. For the Suliciu model, which arises from the numerical approximation of isentropic flows, the semilinear behavior is obtained using pointwise estimates of the gradient. For the Kerr-Debye systems, which arise in nonlinear optics, we show the semilinear behavior via energy methods. For the original Kerr-Debye model, thanks to the special form of the interaction terms, we can show the global existence of smooth solutions.     

Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data

The Cauchy problem for semilinear heat equations with singular initial data is studied, where N2, >0 is a parameter, and a0, a0. We show that when p>(N+2)/N and (N–2)p<N+2, there exists a positive constant such that the problem has two positive self-similar solutions and with if and no positive self-similar solutions if . Furthermore, for each fixed and in L(R^N) as 0, where w₀ is a non-unique solution to the problem with zero initial data, which is constructed by Haraux and Weissler.Mathematics Subject Classification (2000): 35K55, 35J60     

Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

This paper is devoted to strictly hyperbolic systems and equations with non‐smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions).