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We consider an initial boundary value problem for the equation $u_{t t} - Δ u - \nabla ϕ \cdot \nabla u + f (u) + g (u_{t}) = 0$ . We first prove local and global existence results under suitable conditions on f and g. Then we show that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. This result improves and includes earlier decay results established by the authors.

Decay of positive waves in nonlinear systems of conservation laws

Alberto Bressan, Rinaldo M. Colombo (1998)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Decaying positive entire solutions of the $p$ -Laplacian

Yin Xi Huang (1995)

Czechoslovak Mathematical Journal

Derivation of the Zakharov equations

Benjamin Texier (2005)

Journées Équations aux dérivées partielles

Differential inequalities for a nonlinear partial differential equation of the fourth order

Věra Radochová (1977)

Czechoslovak Mathematical Journal

Dirichlet problem for parabolic equations with continuous coefficients

Julio E. Bouillet (1976)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Dispersive and Strichartz estimates for the wave equation in domains with boundary

Oana Ivanovici (2010)

Journées Équations aux dérivées partielles

In this note we consider a strictly convex domain $Ω \subset ℝ^{d}$ of dimension $d \geq 2$ with smooth boundary $\partial Ω \neq \emptyset$ and we describe the dispersive and Strichartz estimates for the wave equation with the Dirichlet boundary condition. We obtain counterexamples to the optimal Strichartz estimates of the flat case; we also discuss the some results concerning the dispersive estimates.

Dispersive estimates and absence of embedded eigenvalues

Herbert Koch, Daniel Tataru (2005)

Journées Équations aux dérivées partielles

In [2] Kenig, Ruiz and Sogge proved ${∥ u ∥}_{L^{\frac{2 n}{n - 2}} (ℝ^{n})} ≲ {∥ L u ∥}_{L^{\frac{2 n}{n + 2}} (ℝ^{n})}$ provided $n \geq 3$ , $u \in C_{0}^{\infty} (ℝ^{n})$ and $L$ is a second order operator with constant coefficients such that the second order coefficients are real and nonsingular. As a consequence of [3] we state local versions of this inequality for operators with $C^{2}$ coefficients. In this paper we show how to apply these local versions to the absence of embedded eigenvalues for potentials in $L^{\frac{n + 1}{2}}$ and variants thereof.

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Das Ausstrahlungsproblem für elliptische Differentialgleichungen in Gebieten mit unbeschränkten Rand.

Das Rand-Anfangswertproblem für quasilineare Wellengleichungen in Sobolevräumen niedriger Ordnung.

Das Verhalten der Ableitungen globaler Lösungen nicht-linearer Wellengleichungen für grosse Zeiten.

Decay estimates of heat transfer to melton polymer flow in pipes with viscous dissipation.

Decay estimates of solutions of a nonlinearly damped semilinear wave equation

Decay of positive waves in nonlinear systems of conservation laws

Decaying positive entire solutions of the $p$ -Laplacian

Derivation of the Zakharov equations

Differential inequalities for a nonlinear partial differential equation of the fourth order

Dirichlet problem for parabolic equations with continuous coefficients

Dispersive and Strichartz estimates for the wave equation in domains with boundary

Dispersive estimates and absence of embedded eigenvalues

Dispersive estimates for a linear wave equation with electromagnetic potential.

Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de $R^{n}$

Dual variational principles for an elliptic partial differential equation

Dynamic Theory of Quasilinear Parbolic Systems. III. Global Existence.

Currently displaying 1 – 16 of 16