Strongly damped wave equations with critical nonlinearities

This note deals with the strongly damped nonlinear wave equation

_utt−Δ_ut−Δu+f(_ut)+g(u)=h$u_{tt}-\Delta u_t-\Delta u+f\left(u_t\right)+g\left(u\right)=h$

with Dirichlet boundary conditions, where both the nonlinearities f$f$ and g$g$ exhibit a critical growth, while h$h$ is a time-independent forcing term. The existence of an exponential attractor of optimal regularity is proven. As a corollary, a regular global attractor of finite fractal dimension is obtained.     

Bifurcation of periodic solutions for wave equations with scalar nonlinearities

We treat wave equations with “scalar nonlinearities” and demonstrate the connection between the bifurcation theory of an associated system of Hammerstein integral equations and the existence of periodic solutions of the nonlinear wave equation.     

On semilinear elliptic equations with indefinite nonlinearities

This paper concerns semilinear elliptic equations whose nonlinear term has the formW(x)f(u) whereW changes sign. We study the existence of positive solutions and their multiplicity. The important role played by the negative part ofW is contained in a condition which is shown to be necessary for homogeneousf. More general existence questions are also discussed.Supported in part by NSF grant DMS9003149.     

Partial differential equations involving subcritical,critical and supercritical nonlinearities

We establish the existence of at least one nonnegative solution for the problem$\text{(}\text{P}\text{)}{}_{\mathrm{\lambda }}\text{−div(a(|}\text{∇}\text{u|)}\text{∇}\text{u)}\text{=}\text{λf(u)}\text{in}\text{Ω,}\text{u}\text{=}\text{0}\text{on}\text{∂Ω,}$where a and f satisfy conditions near zero. Here the novelty is that we do not need restrictions on the nonlinearities at infinity. Therefore, we can consider subcritical, critical and supercritical cases.     

An existence and uniqueness theorem for wave equations with scalar nonlinearities

We prove an existence and uniqueness theorem of global solutions for wave equations with scalar nonlinearities. Our paper is a generalization of the work of R. W. Dickey [6].

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Zusammenfassung Es wird ein Existenz- und Eindeutigkeitssatz für globale Lösungen von Wellengleichungen mit skalaren Nichtlinearitäten bewiesen. Die Arbeit stellt eine Verallgemeinerung der Arbeit [6] von R. W. Dickey dar.

     

Existence and uniqueness of global solutions for wave equations with scalar nonlinearities

We prove an existence and uniqueness theorem of global solutions for wave equations with scalar nonlinearities. Our paper is a generalization of the work of D. Kremer [4].

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Zusammenfassung Es wird ein Existenz- und Eindeutigkeitssatz für globale Lösungen von Wellengleichungen mit skalaren Nichtlinearitäten bewiesen. Die Arbeit stellt eine Verallgemeinerung der Arbeit von D. Kremer [4] dar.

     

Nonlinear equations with expansive nonlinearities

Summary We prove existence and multiplicity theorems for nonlinear equations at resonance with expansive nonlinearities.

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Riassunto Si provano teoremi di esistenza e molteplicità per equazioni nonlineari in risonanza con nonlinearità espansiva.

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These results were obtained while the second author was visiting the University of Ferrara through a grant of C.N.R.

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Supported by C.N.R., G.N.A.F.A.     

On product nonlinearities in stochastic differential equations

The author's methods for solution of stochastic differential equations shown elsewhere to be applicable for nonlinear cases including polynomial, exponential, and trigonometric terms, is shown to be applicable to product nonlinearities as well.     

Global existence for systems of wave equations with nonresonant nonlinearities and null forms

We consider the Cauchy problem for systems of nonlinear wave equations with different propagation speeds in three space dimensions. We prove global existence of small amplitude solutions for systems with some nonresonant nonlinearities which may depend on both of the unknowns and their derivatives. Our method here can be also adopted to treat the null forms.     

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).     

Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping

We consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources); (2) show that the corresponding solutions satisfy Hadamard well-posedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping.     

Evolution differential equations with non-Lipschitz nonlinearities

We suggest a generalization of the Peano theorem (on the existence of a solution of the Cauchy problem for a differential system) to the case of an infinite-dimensional phase space.