Blow-up Result Research Articles

Blow-up theorems for a structural acoustics model

This article studies the finite time blow-up of weak solutions to a structural acoustics model consisting of a semilinear wave equation defined on a bounded domain Ω⊂R3 which is strongly coupled with a Berger plate equation acting on the elastic wall, namely, a flat portion of the boundary. The system is influenced by several competing forces, including boundary and interior source and damping terms. We stress that the power-type source term acting on the wave equation is allowed to have a supercritical exponent, in the sense that its associated Nemytskii operator is not locally Lipschitz from H1 into L2. In this paper, we prove the blow-up results for weak solutions when the source terms are stronger than damping terms, by considering two scenarios of the initial data: (i) the initial total energy is negative; (ii) the initial total energy is positive but small, while the initial quadratic energy is sufficiently large. The most significant challenge in this work arises from the coupling of the wave and plate equations on the elastic wall.

Remarks on blowup of solutions for one‐dimensional compressible Navier–Stokes equations with Maxwell's law

AbstractIn this note, we present some blowup results of solutions to the one‐dimensional compressible Navier–Stokes equations with Maxwell's law. First, we improve the blowup result of Hu and Wang [Math. Nachr. 92 (2019), 826–840] with initial density away from vacuum by removing two restrictions. Next, we give a blowup result for the solutions with decay at far fields. Finally, we construct some special analytical solutions to exhibit the blowup or non‐blowup phenomena for the relaxed system.

Local Solvability, Blow-Up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: III. Cauchy Problems

Three Cauchy problems for Sobolev-type equations with a common linear part from the theory of ion acoustic and drift waves in a plasma are considered. The problems are reduced to equivalent integral equations. We prove the existence of unextendable solutions for two problems and the existence of a local-in-time solution for the third problem. For one of the problems, by applying a modified method of Kh.A. Levin, sufficient conditions for finite time blow-up of solutions are obtained and an upper bound for the solution blow-up time is found. For another problem, S.I. Pohozaev’s nonlinear capacity method is used to obtain a finite time blow-up result and two results concerning the nonexistence of even local solutions, and an upper bound for the solution blow-up time is obtained as well.

General decay and lifespan of solutions for a system of nonlinear pseudoparabolic equations with viscoelastic term

In this paper, we consider the Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with viscoelastic term. By the Faedo-Galerkin method, we first prove the existence and uniqueness of solution for the problem. Next, we give a sufficient condition to get the global existence and decay of the weak solution. Finally, by the concavity method, we prove the blow-up result of the solution when the initial energy is negative. Furthermore, we establish here the lifespan of the solution by finding the upper bound and the lower bound for the blow-up time.

The Exponential Growth of Solution, Upper and Lower Bounds for the Blow-Up Time for a Viscoelastic Wave Equation with Variable- Exponent Nonlinearities

This paper aims to study the model of a nonlinear viscoelastic wave equation with damping and source terms involving variable-exponent nonlinearities. First, we prove that the energy grows exponentially, and thus in 𝐿p2 and 𝐿p1 norms. For the case 2 ≤ 𝑘(. ) < 𝑝(. ), we reach the exponential growth result of a blowup in finite time with positive initial energy and get the upper bound for the blow-up time. For the case 𝑘(. ) = 2, we use the concavity method to show a finite time blow-up result and get the upper bound for the blow-up time. Furthermore, for the case 𝑘(. ) ≥ 2, under some conditions on the data, we give a lower bound for the blow-up time when the blow-up occurs.

Dynamics properties for a viscoelastic Kirchhoff-type equation with nonlinear boundary damping and source terms

This work is devoted to studying a viscoelastic Kirchhoff-type equation with nonlinear boundary damping-source interactions in a bounded domain. Under suitable assumptions on the kernel function g, density function, and M, the global existence and general decay rate of solution are established. Moreover, we prove the finite time blow-up result of solution with negative initial energy.

Scale-invariant semilinear damped wave models with mass term and integrable in time speed of propagation

In this paper, we consider the Cauchy problem for scale-invariant semilinear wave models with time-dependent mass, dissipation and integrable time-dependent propagation speed. The goal is to study the interplay between the coefficients appearing in the mass and dissipation term and the exponent in the speed of propagation to prove global (in time) existence of small data Sobolev solutions and blow-up results.

On the critical curve for systems of hyperbolic inequalities in an exterior domain of the half-space

We establish blow-up results for a system of semilinear hyperbolic inequalities in an exterior domain of the half-space. The considered system is investigated under an inhomogeneous Dirichlet-type boundary condition depending on both time and space variables. In certain cases, an optimal criterium of Fujita-type is derived. Our results yield naturally sharp nonexistence criteria for the corresponding stationary wave system and equation.

Blow-up for a non-linear stable non-Gaussian process in fractional time

The behaviour of solutions for a non-linear diffusion problem is studied. A subordination principle is applied to obtain the variation of parameters formula in the sense of Volterra equations, which leads to the integral representation of a solution in terms of the fundamental solutions. This representation, the so-called mild solution, is used to investigate some properties about continuity and non-negativeness of solutions as well as to prove a Fujita type blow-up result. Fujita’s critical exponent is established in terms of the parameters of the stable non-Gaussian process and a result for global solutions is given.

Blow-up for compressible Euler system with space-dependent damping in 1-D

AbstractThis article considers the Cauchy problem for compressible Euler system inR{\bf{R}}with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs toL1(R){L}^{1}\left({\bf{R}})), then the damping is non-effective to the long-time behavior of the solution.

General Decay and Blow-up Results of a Robin-Dirichlet Problem for a Pseudoparabolic Nonlinear Equation of Kirchhoff-Carrier Type with Viscoelastic Term

General Decay and Blow-up Results of a Robin-Dirichlet Problem for a Pseudoparabolic Nonlinear Equation of Kirchhoff-Carrier Type with Viscoelastic Term

Notes on wave-breaking phenomena for a Fornberg-Whitham-type equation

We investigate the wave-breaking mechanism of solutions to a Fornberg-Whitham-type equation. By means of a weaker conserved L2-norm, we show some wave-breaking criteria for the equation, which improve some blow-up results on the classical Fornberg-Whitham equation. Moreover, our conclusions suggest that the wave breaking for the equation may occur even with small slope of the initial value. To analyze the interaction between nonlinear and nonlocal dispersion terms, we provide two different approaches. The first one is based on a subtle analysis on evolution of the solution u and its gradient ux. The other is to make full use of the known blow-up results on Riccati-type inequalities with t-dependent functions.

Properties of Weak Solutions for a Pseudoparabolic Equation with Logarithmic Nonlinearity of Variable Exponents

In this paper, a new pseudoparabolic equation with logarithmic nonlinearity of variable exponents is investigated. By using the energy functional and the classical potential well, we obtain the global existence and blow-up results of weak solutions with variable exponents. These results extend and improve some recent results in which the blow-up results were showed involving the logarithmic nonlinearity with variable exponents.

A New Blow-Up Criterion to a Singular Non-Newton Polytropic Filtration Equation

In this paper, a singular non-Newton polytropic filtration equation under the initial-boundary value condition is revisited. The finite time blow-up results were discussed when the initial energy E(u0) was subcritical (E(u0)<d), critical (E(u0)=d), and supercritical (E(u0)>d), with d being the potential depth by using the potential well method and some differential inequalities. The goal of this paper is to give a finite time blow-up result if E(u0) is independent of d. Moreover, the explicit upper bound of the blow-up time is obtained by the classical Levine’s concavity method, and the precise lower bound of the blow-up time is derived by applying an interpolation inequality.

Blow-Up Solutions for the Space-Time Fractional Evolution Equation

This paper focuses on the blow-up solutions of the space-time fractional equations with Riemann–Liouville type nonlinearity in arbitrary-dimensional space. Using the Banach mapping principle and the test function method, we establish the local well-posedness and overcome the difficulties caused by the fractional operators to obtain the blow-up results. Furthermore, we get the precise lifespan of blow-up solutions under special initial conditions.

Blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations

In this paper, we study the blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations of radiation hydrodynamics. First, we improve the blowup result in [P. Jiang and Y. G. Wang, Initial-boundary value problems and formation of singularities for one-dimensional non-relativistic radiation hydrodynamic equations, J. Hyperbolic Differential Equations 9 (2012) 711–738] on the half line [Formula: see text] for large initial data by removing a restrict condition. Next, we obtain a new blowup result on the half line [Formula: see text] by introducing a new momentum weight. Finally, we present two non-global existence results for the smooth solutions to the one-dimensional Euler–Boltzmann equations with vacuum on the interval [Formula: see text] by introducing some new average quantities.

Blow-up and lifespan estimates for solutions to the weakly coupled system of nonlinear damped wave equations outside a ball

In this paper, we consider the initial-boundary value problems with several fundamental boundary conditions (the Dirichlet/Neumann/Robin boundary condition) for the multi-component system of semi-linear classical damped wave equations outside a ball. By applying a test function approach with a judicious choice of test functions, which approximates the harmonic functions being subject to these boundary conditions on $$\partial \varOmega $$ , simultaneously we have succeeded in proving the blow-up result in a finite time as well as in catching the upper bound of lifespan estimates for small solutions in all spatial dimensions. Moreover, such kind of these results, which become sharp in the subcritical cases for one-dimensional case, will be discussed at the end of this paper.

Blow-up for the coupled system of wave equations with memory terms in Schwarzschild spacetime

This paper is concerned with blow-up results of solutions to the Cauchy problem for a coupled system of wave equations in Schwarzschild spacetime, where the nonlinear terms contain power-type memory terms, derivative-type memory terms, and combined-type memory terms, respectively. Upper-bound lifespan estimates of solutions to the Cauchy problem with small initial values and certain assumptions on the exponents in the memory terms are established by taking advantage of the Regge–Wheeler coordinate and iteration approach. Our main new contribution is that lifespan estimates of solutions to the problem are associated with the well-known Strauss conjecture and Glassey conjecture in Schwarzschild spacetime. To the best of the authors’ knowledge, the results in Theorems 1.1–1.3 are new.

Existence and blow up time estimate for a nonlinear Cauchy problem with variable exponents: theory and numerics

A nonlinear Cauchy problem of a wave equation with variable exponents is considered. The damping term is subjected to a coefficient of both variables space and time. We prove the local existence of solutions using the Galerkin approach while the blow-up result is obtained via the perturbed energy method. We finally end our paper with some numerical illustrations. Problems that are frequently used in the literature are special cases.

Curvature blow-up and peakons for the Camassa–Holm type equation

In this paper we are concerned with a periodic Camassa–Holm type equation, which includes several integrable systems as particular cases such as the Camassa–Holm equation, the Novikov equation and the modified Camassa–Holm equation. By applying the method of characteristics, the blow-up criterion and the precise blow-up scenario are established. Then the curvature blow-up result is presented. Finally, periodic peakons are expressed in an explicit function formula.