Lifespan Of Solutions Research Articles

On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori

We consider a completely resonant nonlinear Schrödinger equation on the d-dimensional torus, for any d≥1, with polynomial nonlinearity of any degree 2p+1, p≥1, which is gauge and translation invariant. We study the behaviour of high Sobolev Hs-norms of solutions, s≥s1+1>d/2+2, whose initial datum u0∈Hs satisfies an appropriate smallness condition on its lowHs1 and L2-norms respectively. We prove a polynomial upper bound on the possible growth of the Sobolev norm Hs over finite but long time scale that is exponential in the regularity parameter s1. As a byproduct we get stability of the low Hs1-norm over such time interval. A key ingredient in the proof is the introduction of a suitable “modified energy” that provides an a priori upper bound on the growth. This is obtained by combining para-differential techniques and suitable tame estimates.

On the Effect of Rotation on the Life-Span of Analytic Solutions to the 3D Inviscid Primitive Equations

We study the effect of the rotation on the life-span of solutions to the 3D hydrostatic Euler equations with rotation and the inviscid Primitive equations (PEs) on the torus. The space of analytic functions appears to be the natural space to study the initial value problem for the inviscid PEs with general initial data, as they have been recently shown to exhibit Kelvin–Helmholtz type instability. First, for a short interval of time that is independent of the rate of rotation $$|\Omega |$$ , we establish the local well-posedness of the inviscid PEs in the space of analytic functions. In addition, thanks to a fine analysis of the barotropic and baroclinic modes decomposition, we establish two results about the long time existence of solutions. (i) Independently of $$|\Omega |$$ , we show that the life-span of the solution tends to infinity as the analytic norm of the initial baroclinic mode goes to zero. Moreover, we show in this case that the solution of the 3D inviscid PEs converges to the solution of the limit system, which is governed by the 2D Euler equations. (ii) We show that the life-span of the solution can be prolonged unboundedly with $$|\Omega |\rightarrow \infty $$ , which is the main result of this paper. This is established for “well-prepared” initial data, namely, when only the Sobolev norm (but not the analytic norm) of the baroclinic mode is small enough, depending on $$|\Omega |$$ . Furthermore, for large $$|\Omega |$$ and “well-prepared” initial data, we show that the solution to the 3D inviscid PEs is approximated by the solution to a simple limit resonant system with the same initial data.

Global existence, nonexistence, and decay of solutions for a viscoelastic wave equation with nonlinear boundary damping and source terms

This paper deals with a high order viscoelastic wave equation with nonlinear boundary damping–source interactions. By the combination of Galerkin approximation and potential well methods, we obtain the global existence of weak solutions. Under some appropriate assumptions on the boundary damping–source terms and the relaxation function g, we establish general decay and blow-up results associated with solution energy. Estimates of the lifespan of solutions are also given.

Blow-up and Lifespan of Solutions for a Nonlinear Viscoelastic Kirchhoff Equation

The blow-up of solutions of a class of nonlinear viscoelastic Kirchhoff equation with suitable initial data and Dirichlet boundary conditions is discussed. By constructing a suitable auxiliary function to overcome the difficulty of gradient estimation and making use of differential inequality technique, we establish a finite time blow-up result when the initial data is at arbitrary energy level. Moreover, a lower bound of the lifespan is also derived by constructing a control function with both nonlocal term and memory kernel. Compared with the previous literature, our approach to estimate the lifespan does not require the initial energy to control some norms of the solution.

Lifespan of solutions to semilinear damping wave equations in de Sitter spacetime

This paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de Sitter spacetime. The first problem comes from the latter problem of conjecture of Strauss in de Sitter Spacetime, and the second problem arises from the latter problem of conjecture of Glassey in de Sitter Spacetime. We give blowup and lifespan of solutions on those two problems. Comparing with non-damping case established in Yan (2018), one can see that the damping term with a positive bounded time-dependent coefficient does not effect blowup of solutions. Meanwhile, we establish a new ODE blowup result concerning damping term.

Life span of solutions with large initial data for a semilinear parabolic system coupling exponential reaction terms

In this paper, we study a coupled systems of parabolic equations subject to large initial data. By using comparison principle and Kaplan’s method, we get the upper and lower bound for the life span of the solutions.

Remarks on the lifespan ofsolutions to wave equationswith a potential

In this paper, we show that the solution to the semilinear wave equation with a potential of quadratic type blows up in finite time. We also give an upper bound estimate for the lifespan of the solution.

The lifespan of 3D radial solutions to the non-isentropic relativistic Euler equations

This paper investigates the lower bound of the lifespan of three-dimensional spherically symmetric solutions to the non-isentropic relativistic Euler equations, when the initial data are prescribed as a small perturbation with compact support to a constant state. Based on the structure of the hyperbolic system, we show the almost global existence of the smooth solutions to Eulerian flows (polytropic gases and generalized Chaplygin gases) with genuinely nonlinear characteristics. While for the Eulerian flows (Chaplygin gas and stiff matter) with mild linearly degenerate characteristics, we show the global existence of the radial solutions, moreover, for the non-strictly hyperbolic system (pressureless perfect fluid) satisfying the mild linearly degenerate condition, we prove the blowup phenomenon of the radial solutions and show that the lifespan of the solutions is of order $$O(\epsilon ^{-1})$$ , where $$\epsilon $$ denotes the width of the perturbation. This work can be seen as a complement of our work (Lei and Wei in Math Ann 367:1363–1401, 2017) for relativistic Chaplygin gas and can also be seen as a generalization of the classical Eulerian fluids (Godin in Arch Ration Mech Anal 177:497–511, 2005, J Math Pures Appl 87:91–117, 2007) to the relativistic Eulerian fluids.

Blow up of Solutions to a Class of Damped Viscoelastic Inverse Source Problem

This article is devoted to the study blow up of solutions to a quasilinear inverse source problem with memory and damping terms. We obtain sufficient conditions on initial functions for which the solutions blow up in a finite time. Estimates of the lifespan of solutions are also given.

Estimates of blow‐up times of a system of semilinear SPDEs

In this paper, blow‐up property to a system of nonlinear stochastic PDEs driven by two‐dimensional Brownian motions is investigated. The lower and upper bounds for blow‐up times are obtained. When the system parameters satisfy certain conditions, the explicit solutions of a related system of random PDEs are deduced, which allows us to use Yor's formula to obtain the distribution functions of several blow‐up times. Particularly, the impact of noises on the life span of solutions is studied as the system parameters satisfy different conditions. Copyright © 2016 John Wiley & Sons, Ltd.

Secondary critical exponent and life span for a doubly singular parabolic equation with a weighted source

This paper deals with the Cauchy problem for a doubly singular parabolic equation with a weighted source ut=div(|∇u|p−2∇um)+|x|αuq, (x,t)∈ℝN×(0,T),where N ≤ 1, 1 <p < 2, m>max{0,3−p−pN} satisfying 2 <p + m < 3, q > 1, and α >N(3 – p – m) – p. We give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and non-existence of global solutions of the Cauchy problem. Moreover, the life span of solutions is also studied.

The Cauchy problem for the homogeneous Monge–Ampère equation, III. Lifespan

Abstract We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge–Ampère equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the C 3 ${C^{3}}$ lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and intersection of complex time characteristics. We use a conservation law type argument to prove uniqueness of solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy problem is ill-posed in C 3 ${C^{3}}$ , in the sense that there exists a dense set of C 3 ${C^{3}}$ Cauchy data for which there exists no C 3 ${C^{3}}$ solution even for a short time. In the real domain we show that the HRMA is equivalent to a Hamilton–Jacobi equation, and use the equivalence to prove that any differentiable weak solution is smooth, so that the differentiable lifespan equals the convex lifespan determined in our previous articles. We further show that the only obstruction to C 1 ${C^{1}}$ solvability is the invertibility of the associated Moser maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a positive but generally finite time and cannot be continued even as a weak C 1 ${C^{1}}$ solution afterwards. Finally, we introduce the notion of a “leafwise subsolution” for the HCMA that generalizes that of a solution, and many of our aforementioned results are proved for this more general object.

Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations

In this paper, a blow-up problem to nonlinear stochastic partial differential equations driven by Brownian motions is investigated. In particular, the impact of noises on the life span of solutions is studied. It is interesting to know that the noise can be used to postpone the blow-up time of a stochastic nonlinear system.

Life span and a new critical exponent for a doubly degenerate parabolic equation with slow decay initial values

In this paper, we investigate the behavior of the positive solution of the following Cauchy problem u t − div ( | ∇ u m | p − 2 ∇ u m ) = u q with initial value decaying at infinity, and give a new secondary critical exponent for the existence of global and nonglobal solutions. Furthermore, the large time behavior and the life span of solutions are also studied.

The lifespan for 3D quasilinear wave equations with nonlinear damping terms

In this paper, we study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain Ω . The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results in [K. Agre, M.A. Rammaha, System of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006) 1235–1270], especially, the blow-up of weak solutions in the case of non-negative energy.

BLOW-UP OF POSITIVE-INITIAL-ENERGY SOLUTIONS FOR AN INTEGRO-DIFFERENTIAL EQUATION WITH NONLINEAR DAMPING

The initial boundary value problem for an integro-differential equation with nonlinear damping in a bounded domain is considered. The local existence and blow-up of solutions with positive initial energy are discussed under some conditions. Estimates of the lifespan of solutions are also given.

Blow-Up Results for a Nonlinear Hyperbolic Equation with Lewis Function

The initial boundary value problem for a nonlinear hyperbolic equation with Lewis function in a bounded domain is considered. In this work, the main result is that the solution blows up in finite time if the initial data possesses suitable positive energy. Moreover, the estimates of the lifespan of solutions are also given.

Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values

In this paper, we consider the positive solution of a Cauchy problem for the following P -Laplace parabolic equation u t = div ( | ∇ u | p − 2 ∇ u ) + u q , p > 2 , q > 1 , and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence of global and nonglobal solutions of the Cauchy problem. Furthermore, the life span of solutions is also studied.

Life span of solutions with large initial data for a superlinear heat equation

We investigate the initial-boundary problem { u t = Δ u + f ( u ) in Ω × ( 0 , ∞ ) , u = 0 on ∂ Ω × ( 0 , ∞ ) , u ( x , 0 ) = ρ φ ( x ) in Ω , where Ω is a bounded domain in R N with a smooth boundary ∂ Ω, ρ > 0 , φ ( x ) is a nonnegative continuous function on Ω ¯ , f ( u ) is a nonnegative superlinear continuous function on [ 0 , ∞ ) . We show that the life span (or blow-up time) of the solution of this problem, denoted by T ( ρ ) , satisfies T ( ρ ) = ∫ ρ ‖ φ ‖ ∞ ∞ d u f ( u ) + h.o.t. as ρ → ∞ . Moreover, when the maximum of φ is attained at a finite number of points in Ω, we can determine the higher-order term of T ( ρ ) which depends on the minimal value of | Δ φ | at the maximal points of φ. The proof is based on a careful construction of a supersolution and a subsolution.

Life span of solutions of a strongly coupled parabolic system

Life span of solutions of a strongly coupled parabolic system