Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

Global existence of solutions of the critical semilinear wave equations with variable coefficients outside obstacles

In this paper, we consider the exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove the global existence of smooth solutions. As in the constant coefficients case, we show that the energy cannot concentrate at any point (t, x) ∈ (0, ∞) ×Ω. For that purpose, following Ibrahim and Majdoub's paper in 2003, we use a geometric multiplier similar to the well-known Morawetz multiplier used in the constant coefficients case. We then use the compari...     

Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part

We consider the equation ${\mathrm{\Delta }}_{g}u+hu=|u{|}^{2^{?}?2}u$ in a closed Riemannian manifold $(M,g)$, where $h\in C^{0,\theta }(M)$, $\theta \in (0,1)$ and $2^{?}=\frac{2n}{n?2}$, $n:=\dim ?(M)\ge 3$. We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is a priori bounded. We obtain this result under the conditions that $n\ge 7$ and $h<\frac{n?2}{4(n?1)}{\text{Scal}}_g$ in M, where ${\text{Scal}}_g$ is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function h.     

Three sphere inequality for second order elliptic equations with coefficients with jump discontinuity

This is a short note to complete the paper appeared in Francini et al. (2016) [4], where a rough version of the classical well known Hadamard three-circle theorem for solution of an elliptic PDE in divergence form has been proved. Precisely, instead of circles, the authors obtain a similar inequality in a more complicated geometry. In this paper we clean the geometry and obtain a generalized version of the three-circle inequality for elliptic equation with coefficients with discontinuity of jump type.     

Entropy solutions for stochastic porous media equations

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\mathrm{\Delta }(|u{|}^{m?1}u)$ for all $m\in (1,\infty )$, and Hölder continuous diffusion nonlinearity with exponent 1/2.     

Blow-up of solutions of semilinear parabolic differential equations

For a semilinear parabolic initial boundary value problem we establish criterions on blow-up of the solution in finite time and give bounds for the blow-up time. We treat several applications in both finite and infinite domains. For comparison, sufficient conditions are also given for the existence of global solutions.

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Zusammenfassung Für ein semilineares parabolisches Rand- und Anfangswertproblem stellen wir Kriterien für die Explosion der Lösung in endlicher Zeit auf und geben Schranken für die Explosionszeit an. Einige Anwendungen in beschränkten und unbeschränkten Gebieten werden untersucht, wobei wir als Gegenüberstellung auch hinreichende Bedingungen für die Existenz globaler Lösungen angeben.

     

Singular solutions of elliptic equations on a perturbed cone

In this work we obtain positive singular solutions of

$\{\begin{array}{ccc}?\mathrm{\Delta }u(y)\hfill & \hfill =\hfill & u{(y)}^p\text{ in }y\in {\mathrm{\Omega }}_t,\hfill \\ u\hfill & \hfill =\hfill & 0\text{ on }y\in ?{\mathrm{\Omega }}_t,\hfill \end{array}$

where ${\mathrm{\Omega }}_t$ is a sufficiently small $C^{2,\alpha }$ perturbation of the cone $\mathrm{\Omega }:=\{x\in {\mathbb{R}}^N:x=r\theta ,r>0,\theta \in S\}$ where $S?S^{N?1}$ has a smooth nonempty boundary and where $p>1$ satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone Ω and here we use a different class of function spaces.     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients

In the present paper, using a Picard Type method of approximation, we investigate the global existence of mild solutions for a class of Ito Type stochastic differential equations whose coefficients satisfy conditions more general than the Lipschitz and linear growth ones.     

Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball

In this article we obtain positive singular solutions of

(1)$?\mathrm{\Delta }u=|\mathrm{?}u{|}^p\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a small $C^2$ perturbation of the unit ball in ${\mathbb{R}}^N$. For $\frac{N}{N?1}<p<2$ we prove that if Ω is a sufficiently small $C^2$ perturbation of the unit ball there exists a singular positive weak solution u of (1). In the case of $p>2$ we prove a similar result but now the positive weak solution u is contained in $C^{0,\frac{p?2}{p?1}}(\stackrel{￣}{\mathrm{\Omega }})$ and yet is not in $C^{0,\frac{p?2}{p?1}+\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ for any $\epsilon >0$.     

Boundary stabilization of wave equations with variable coefficients

The aim of this paper is to obtain the exponential energy decay of the solution of the wave equation with variable coefficients under suitable linear boundary feedback. Multiplier method and Riemannian geometry method are used.     

Blow-up of solutions of some nonlinear wave equations

We analyze the problem of blow-up of global solutions of a semilinear wave equation with a potential and with a possible degeneration at infinity for nonnegative initial data with compact support. By using the nonlinear capacity method, we prove the theorem on the nonexistence of such a solution for a subcritical and the critical nonlinearity exponent.     

Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities

In this paper we discuss continuation properties and asymptotic behavior of -regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions. A necessary and sufficient condition for global in time existence of -regular solutions is given. We also formulate sufficient conditions to construct a piecewise -regular solutions (continuation beyond maximal time of existence for -regular solutions). Applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed. In particular global solvability and the existence of a global attractor for in is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition. Similar result is obtained for a 2mth order semilinear parabolic initial boundary value problem in a Hilbert space .     

Long-time solutions of scalar nonlinear hyperbolic reaction equations incorporating relaxation I. The reaction function is a bistable cubic polynomial

We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general form

$u_{\tau \tau }+u_{\tau }=u_{xx}+\epsilon (F(u)+F{(u)}_{\tau }),$

in which x and τ represent dimensionless distance and time respectively and $\epsilon >0$ is a parameter related to the relaxation time. Furthermore the reaction function, $F(u)$, is given by the bistable cubic polynomial,

$F(u)=u(1?u)(u?\mu ),$

in which $0<\mu <1/2$ is a parameter. The initial data is given by a simple step function with $u(x,0)=1$ for $x\le 0$ and $u(x,0)=0$ for $x>0$. It is established, via the method of matched asymptotic expansions, that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wave front which is either of reaction–diffusion or of reaction–relaxation type. The one exception to this occurs when $\mu =\frac{1}{2}$ in which case the large time attractor for the solution of the initial-value problem is a stationary state solution of kink type centred at the origin.     

Positive solutions of semilinear biharmonic equations with critical Sobolev exponents

In this paper we study the critical growth biharmonic problem with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the biharmonic problem.