Blow Up of Solutions to the Cauchy Problem for Nonlinear Wave Equations

Zhou, Yi

doi:10.1142/s0252959901000280

Cited by 109 publications

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“…where K 0 n−1 2 α, and a, b are positive constants depending only on α and n. As stated in [5], all lower bounds in (1.2), except the case n = 2 and α = 2, are known to be sharp due to Lax [4] (for n = 1 and α = 1), John [1] and Zhou [13] (for n = 2, 3 and α = 1), Kong [3] (for n = 1 and α ≥ 1) and Zhou [13] (for n ≥ 1 and odd α ≥ 1). However, up to now there is no sharpness result on the lower bound of the lifespan…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Zhou [13] and Takamura [12] obtained the blow-up result, and gave the estimate on the lifespan. In particular, when p = 3 and n = 2, (1.5) becomes…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…is a smooth non-negative and radially symmetric function on R 2 with compact support The related studies on the blow-up of solutions to nonlinear wave equations can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

Section: Theorem 11 Suppose That G(x)mentioning

confidence: 99%

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Sharpness on the lower bound of the lifespan of solutions to nonlinear wave equations

Zhou

Han

2011

Chin. Ann. Math. Ser. B

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This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity (see the results of T. T. Li and Y. M. Chen in 1992). For this purpose, the authors consider the following Cauchy problem:∂ 2 x i is the wave operator, g(x) ≡ 0 is a smooth non-negative function on R 2 with compact support, and ε > 0 is a small parameter. It is shown that the solution blows up in a finite time, and the lifespan T (ε) of solutions has an upper bound T (ε) ≤ exp(Aε −2 ) with a positive constant A independent of ε, which belongs to the same kind of the lower bound of the lifespan.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Zhou [13] and Takamura [12] obtained the blow-up result, and gave the estimate on the lifespan. In particular, when p = 3 and n = 2, (1.5) becomes…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Sharpness on the lower bound of the lifespan of solutions to nonlinear wave equations

Zhou

Han

2011

Chin. Ann. Math. Ser. B

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“…In the case of classical C ∞ 0 initial data with size of order , the global or almost global existence can be proved for p ≥ 1+ n−1 ), see Sogge [10]. When p = 1 + 2 n−1 , the lifespan T is also sharp for the problem with nonlinearity |∂ t u| p (Zhou [17]). As in Theorem 1.1, our object here is to prove the corresponding results with low regularity.…”

Section: Discussionmentioning

confidence: 99%

“…Sogge [10]. Moreover, the lifespan T is also sharp for the problem with nonlinearity |∂ t u| 3 (Zhou [17]). Our object here is to prove the corresponding result with low regularity.…”

Section: Introductionmentioning

confidence: 99%

Almost Global Existence for Some Semilinear Wave Equations with Almost Critical Regularity

Fang

Wang

2013

Communications in Partial Differential Equations

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Abstract. For any subcritical index of regularity s > 3/2, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space H s × H s−1 with certain angular regularity. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the spaceIn the last section, we also consider the general semilinear wave equations with the spatial dimension n ≥ 2 and the order of nonlinearity p ≥ 3.

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

Han

2015

Math Methods in App Sciences

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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 onedimensional general quasilinear wave equations u tt u xx D b.u, Du/u xx C 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 (1.1) 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 (1.8) in the general case and (1.10) in the special case.

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Blow Up of Solutions to the Cauchy Problem for Nonlinear Wave Equations

Cited by 109 publications

References 0 publications

Sharpness on the lower bound of the lifespan of solutions to nonlinear wave equations

Sharpness on the lower bound of the lifespan of solutions to nonlinear wave equations

Almost Global Existence for Some Semilinear Wave Equations with Almost Critical Regularity

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

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