Lifespan Of Classical Solutions Research Articles

Life-Span of Classical Solutions to Hyperbolic Inverse Mean Curvature Flow

In this paper, we investigate the life-span of classical solutions to hyperbolic inverse mean curvature flow. Under the condition that the curve can be expressed in the form of a graph, we derive a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system in terms of Riemann invariants. By the theory on the local solution for the Cauchy problem of the quasilinear hyperbolic system, we discuss life-span of classical solutions to the Cauchy problem of hyperbolic inverse mean curvature.

Breakdown of classical solutions to Cauchy problem for inhomogeneous quasilinear hyperbolic systems

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for inhomogeneous quasilinear strictly hyperbolic systems with weaker decaying initial data. Under the assumption that the source term satisfies the corresponding matching condition, we obtains a blow-up result for C1 solution to the Cauchy problem.

The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations utt−uxx=b(u,Du)uxx+F(u,Du). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span in the general case and in the special case. Copyright © 2015 John Wiley & Sons, Ltd.

LOWER BOUND OF THE LIFESPAN OF SOLUTIONS TO SEMILINEAR WAVE EQUATIONS IN AN EXTERIOR DOMAIN

We consider the Cauchy–Dirichlet problem for semilinear wave equations in a three space-dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and Hörmander where the Cauchy problem was treated. We show that our estimate is sharp at least for radially symmetric case.

Life-span of classical solutions to one dimensional initial–boundary value problems for general quasilinear wave equations with Robin boundary conditions

This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations utt−uxx=b(u,Du)uxx+2a0(u,Du)utx+F(u,Du) with Robin boundary conditions on an exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations. Our result in the general case and the special case is shorter than that of the initial–boundary value problem for one dimensional general quasilinear wave equations with Dirichlet boundary conditions. The results in this paper are not the trivial generalization of that in the case of Dirichlet boundary conditions. The lower bound estimates of life span of classical solutions to initial–boundary value problems are consistent with the actual physical meaning. The physical phenomenon also explains our results.

Breakdown of classical solutions to quasilinear hyperbolic systems

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C^1 solution to Cauchy problem.

Lower bounds of the life span of classical solutions to a system of semilinear wave equations in two space dimensions

In this paper we consider a system of semilinear wave equations in two space dimensions and with propagation speeds possibly different from one. Under smallness assumptions on the data, we show lower bounds for the life span of classical solutions.

The blowup for 2-D quasilinear wave equations with cubic nonlinearity

For 2-D quasilinear wave equations with cubic nonlinearity and small initial data, we not only show that the solutions blow up in finite time but also give a complete asymptotic expansion of the lifespan of classical solutions. Hence we solve a problem posed by S. Alinhac and A. Hoshiga. Moreover, as an application of this result, we prove the blowup of solutions for the nonlinear vibrating membrane equations.

LIFESPAN OF SOLUTIONS FOR TWO SPACE DIMENSIONAL WAVE EQUATIONS WITH CUBIC NONLINEARITY

We study the lifespan of classical solutions to the Cauchy problem for nonlinear wave equations of the type with initial data u = εf, u t = ε g at t = 0, where D x = (∂ x 1 , ∂ x 2 ), D = (∂ t , D x ) and ε is a small parameter. Estimates of the lifespan from below are derived for some cases where F is a function of cubic order in a neighborhood of the origin, and ∂3 u F(0, 0, 0) = 0, but not necessarily ∂4 u F(0, 0, 0) = 0.

LIFE-SPAN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH SLOW DECAY INITIAL DATA

The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By constructing an example, first it is illustrated that the classical solution to this kind of Cauchy problem may blow up in a finite time, even if the system is weakly linearly degenerate. Then some lower bounds of the life-span of classical solutions are given in the case that the system is weakly linearly degenerate. These estimates imply that, when the system is weakly linearly degenerate, the classical solution exists almost globally in time. Finally, it is proved that Theorems 1.1–1.3 in [2] are still valid for this kind of initial data.

Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data

Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data

The asymptotic behaviour of solutions for a class of quasilinear wave equations with cubic nonlinearity in two space dimensions

For a class of quasilinear wave equations with small initial data, first we give the lower bound of lifespan of classical solutions, then we discuss the long time asymptotic behaviour of solutions away from the blowup time.

Asymptotic blow-up time of classical solutions for general quasilinear hyperbolic systems and its applications

By means of determining the asymptotic blow-up time, which follows Hörmander (Report, Institute Mittag-Leffler, 1985), for classical solutions to Cauchy problem for quasilinear hyperbolic systems with small decay initial data, we give a limit formula on the life span of classical solutions. Applications to quasilinear hyperbolic systems arising from physics and mechanics are given.

Delayed singularity formation in 2D compressible flow

The initial value problem for the compressible Euler equations in two space dimensions is studied. Of interest is the lifespan of classical solutions with initial data that is a small perturbation from a constant state. The approach taken is to regard the compressible solution as a nonlinear superposition of an underlying incompressible flow and an irrotational compressible flow. This viewpoint yields an improvement for the lifespan over that given by standard existence theory. The estimate for the lifespan is further improved when the initial data possesses certain symmetry. In the case of rotational symmetry, a result of S. Alinhac is reconsidered. The approach is also applied to the study of the incompressible limit. The analysis combines energy and decay estimates based on vector fields related to the natural invariance of the equations.

An elementary proof of the exponential blow‐up for semi‐linear wave equations

AbstractThis paper deals with the upper bound of the life span of classical solutions to □u = ∣u∣p, u∣t = 0 = εφ(x), ut∣t=0 = εψ(x) with the critical power of p in two or three space dimensions. Zhou has proved that the rate of the upper bound of this life span is exp(ε−p(p−1)). But his proof, especially the two‐dimensional case, requires many properties of special functions. Here we shall give simple proofs in each space dimension which are produced by pointwise estimates of the fundamental solution of □. We claim that both proofs are done in almost the same way.

Life-span of classical solutions to fully nonlinear wave equations—II

Life-span of classical solutions to fully nonlinear wave equations—II