Blow-up For Semilinear Wave Equations Research Articles

A competition on blow-up for semilinear wave equations with scale-invariant damping and nonlinear memory term

<p style='text-indent:20px;'>In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, which can be represented by the Riemann-Liouville fractional integral of order <inline-formula><tex-math id="M2">\begin{document}$ 1-\gamma $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \gamma\in(0, 1) $\end{document}</tex-math></inline-formula>. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of <inline-formula><tex-math id="M4">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, on the blow-up range between the effective case and the non-effective case.</p>

Blowup for semilinear wave equation with space-dependent damping and combined nonlinearities

This paper is concerned with the Cauchy problem for semilinear wave equation with space-dependent scattering damping and combined nonlinearities. The blowup results of solution are established by introducing proper test functions. Moreover, upper bound lifespan estimates of a solution to the Cauchy problem with small initial values are derived. To the best of our knowledge, the results in Theorems 1.1–1.2 are new.

Blow-up for semilinear wave equations with time-dependent damping in an exterior domain

<p style='text-indent:20px;'>We consider the semilinear wave equation with time-dependent damping <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_{tt}u-\Delta u +\mu (1+t)^{-\beta} \partial_t u = |u|^p, \quad (t, x)\in (0, \infty)\times D^c, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ D^c = \mathbb{R}^N\backslash D $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ D $\end{document}</tex-math></inline-formula> is the closed unit ball in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\geq 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ -1&lt;\beta&lt;1 $\end{document}</tex-math></inline-formula>. The considered equation is investigated under the boundary conditions: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ u(t, x) \left(\mbox{or } \frac{\partial u}{\partial n^+}(t, x)\right) = b(t)f(x)\, \, \mbox{on}\, \, (0, \infty)\times \partial D, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M8">\begin{document}$ n^+ $\end{document}</tex-math></inline-formula> is the outward (relative to <inline-formula><tex-math id="M9">\begin{document}$ D^c $\end{document}</tex-math></inline-formula>) unit normal on <inline-formula><tex-math id="M10">\begin{document}$ \partial D $\end{document}</tex-math></inline-formula>. General blow-up results are established for the considered problems. Moreover, for a certain class of functions <inline-formula><tex-math id="M11">\begin{document}$ b $\end{document}</tex-math></inline-formula>, the critical exponent in the sense of Fujita is obtained.

Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L2 and Lp+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.

Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent

The blow-up for semilinear wave equations with the scale invariant damping has been well-studied for sub-Fujita exponent. However, for super-Fujita exponent, there is only one blow-up result which is obtained in 2014 by Wakasugi in the case of non-effective damping. In this paper we extend his result in two aspects by showing that: (I) the blow-up will happen for bigger exponent, which is closely related to the Strauss exponent, the critical number for non-damped semilinear wave equations; (II) such a blow-up result is established for a wider range of the constant than the known non-effective one in the damping term.

Finite time blow-up for semilinear wave equations with variable coefficients

Abstract This paper is devoted to studying initial value problems for semilinear wave equations with variable coefficients with subcritical exponents for n (n ≥ 1) space dimensions. We prove that solutions to a certain kind of subcritical wave equations cannot be global if the initial values are positive somewhere and nonnegative no matter how small the initial data are, and also we give the sharp lifespan estimate of solutions for the problems. This solves a part of the famous Strauss' conjecture about semilinear wave equations in the case of subcritical exponents and variable coefficients Cauchy problems.

Blow-up for semilinear wave equation with boundary damping and source terms

In this paper, we consider the semilinear wave equation with boundary damping and source terms. This work is devoted to prove a finite time blow-up result under suitable condition on the initial data and positive initial energy.

On blowup for semilinear wave equations with a focusing nonlinearity

In this paper we report on numerical studies of the formation of singularities for the semilinear wave equations with a focusing power nonlinearity utt − Δu = up in three space dimensions. We show that for generic large initial data that lead to singularities, the spatial pattern of blowup can be described in terms of linearized perturbations about the fundamental self-similar (homogeneous in space) solution. We consider also non-generic initial data which are fine-tuned to the threshold for blowup and identify critical solutions that separate blowup from dispersal for some values of the exponent p.

An elementary proof of the blow-up for semilinear wave equation in high space dimensions

This paper concerns the blow-up of solutions to u tt −Δ u=| u| p in high dimensions for n⩾4 and 1< p< p 0( n), where p 0( n) is a critical exponent. We proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities.

Blow-up for semilinear wave equations in four or five space dimensions

Blow-up for semilinear wave equations in four or five space dimensions

Blow-up for semilinear wave equations with slowly decaying data in high dimensions

We are concerned with classical solutions to the initial value problem for $\square u=|u|^p$ or $\square u=|u_t|^p$ in $\Bbb R^n\times[0,\infty)$ with \lq\lq small" data. If the data have compact support, it is partially known that there is a critical number $p_0(n)$ such that most solutions blow-up in finite time for $1<p\le p_0(n)$ and a global solution exists for $p>p_0(n)$. In this paper, we shall show for all $n\ge2$ that, if the support of data is noncompact, there are blowing-up solutions even for $p>p_0(n)$ because of the \lq\lq bad" spatial decay of the initial data. Moreover, critical decays for each equation are conjectured. The proof lies in the pointwise estimates of the fundamental solution of $\square$.

Blow-up for semilinear wave equations with initial data of slow decay in low space dimensions

Blow-up for semilinear wave equations with initial data of slow decay in low space dimensions