### A note on a theorem of Jörgens.

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En este artículo se estudia el análisis matemático de una ley de conservación que no es clásica. El modelo describe procesos estatigráficos en Geología y tiene en cuenta una condición de tasa de erosión limitada. En primer lugar se presentan el modelo físico y la formulación matemática (posiblemente nueva). Tras enunciar la definición solución se presentan las herramientas que permiten probar la existencia de soluciones.

An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region$\Omega $ (E) ${\left(t{u}_{t}\right)}_{t}={\sum}_{i,j=1}{\left({a}_{ij}\left(x\right){u}_{{x}_{i}}\right)}_{{x}_{j}}-{a}_{0}\left(x\right)u+f\left(u\right)$, subject to the initial and boundary conditions, $u=0$ on $\partial \Omega $ and $u(x,0)={u}_{0}$. (E) is degenerate at $t=0$ and thus, even in the case $f\equiv 0$, time derivatives of $u$ will blow up as $t\to 0$. Also, in the case where $f$ is locally Lipschitz, solutions of (E) can blow up for $t>0$ in finite time. Stability and convergence of the scheme in ${W}^{2,1}$ is shown in the linear case without assuming ${u}_{tt}$ (which can blow up as $t\to 0$ is...

In this paper, we show finite time blow-up of solutions of the p−wave equation in ℝN, with critical Sobolev exponent. Our work extends a result by Galaktionov and Pohozaev [4]

In this paper we prove the ${C}^{\infty}$-well posedness of the Cauchy problem for quasi-linear hyperbolic equations of second order with coefficients non-Lipschitz in t ∈ [0,T] and smooth in x ∈ ℝⁿ.

In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction $$\ddot{u}=-\gamma \dot{u}+{m(\parallel \nabla u\parallel}^{2}{)\Delta u-\delta |u|}^{\alpha}u+f,$$ which is known as degenerate if $m(\xb7)\ge 0$, and non-degenerate if $m(\xb7)\ge {m}_{0}>0$. We would like to point out that, to the author’s knowledge, exponential decay for this type of equations has been studied just for the special cases of $\alpha $. Our aim is to extend the validity of previous results in [5] to $\alpha \ge 0$ both to the degenerate and non-degenerate cases of $m$. We extend our results to equations with...

The present paper studies the existence and uniqueness of global solutions and decay rates to a given nonlinear hyperbolic problem.

The aim of this paper is to prove the existence of the global attractor of the Cauchy problem for a semilinear degenerate damped hyperbolic equation involving the Grushin operator with a locally Lipschitz nonlinearity satisfying a subcritical growth condition.

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $\vartheta $, an evolution equation for the phase change parameter $\chi $, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable $\mathbf{u}$. The main novelty of the model...