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Jun 29, 2018
06/18

by
Kunio Hidano

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In this paper, we show almost global existence of small solutions to the Cauchy problem for symmetric system of wave equations with quadratic (in 3D) or cubic (in 2D) nonlinear terms and multiple propagation speeds. To measure the size of initial data, we employ a weighted Sobolev norm whose regularity index is the smallest among all the admissible Sobolev norms of integer order. We must overcome the difficulty caused by the absence of the $H^1$-$L^p$ Klainerman-Sobolev type inequality, in...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1610.04824

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102

Sep 17, 2013
09/13

by
Kunio Hidano

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This paper is concerned with the global existence of small solutions to pure-power nonlinear Schroedinger equations subject to radially symmetric data with critical regularity. Under radial symmetry we focus our attention on the case where the power of nonlinearity is somewhat smaller than the pseudoconformal power and the initial data belong to the scale-invariant homogeneous Sobolev space. In spite of the negative-order differentiability of initial data the nonlinear Schroedinger equation has...

Source: http://arxiv.org/abs/0711.1941v1

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92

Sep 17, 2013
09/13

by
Kunio Hidano; Yuki Kurokawa

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This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS) inequalities. In the same way we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schroedinger equation. The proof of the weighted HLS inequality for radially symmetric functions involves an...

Source: http://arxiv.org/abs/0711.1933v1

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Jun 27, 2018
06/18

by
Kunio Hidano; Kazuyoshi Yokoyama

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We study the Cauchy problem with small initial data for a system of semilinear wave equations $\square u = |v|^p$, $\square v = |\partial_t u|^p$ in $n$-dimensional space. When $n \geq 2$, we prove that blow-up can occur for arbitrarily small data if $(p, q)$ lies below a curve in $p$-$q$ plane. On the other hand, we show a global existence result for $n=3$ which asserts that a portion of the curve is in fact the borderline between global-in-time existence and finite time blow-up. We also...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1505.05924

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86

Jul 20, 2013
07/13

by
Kunio Hidano; Chengbo Wang; Kazuyoshi Yokoyama

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In this paper, we verify the Glassey conjecture in the radial case for all spatial dimensions, which states that, for the nonlinear wave equations of the form $\Box u=|\nabla u|^p$, the critical exponent to admit global small solutions is given by $p_c=1+\frac{2}{n-1}$. Moreover, we are able to prove the existence results with low regularity assumption on the initial data and extend the solutions to the sharp lifespan. The main idea is to exploit the trace estimates and KSS type estimates.

Source: http://arxiv.org/abs/1107.0847v1

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Jun 30, 2018
06/18

by
Kunio Hidano; Chengbo Wang; Kazuyoshi Yokoyama

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This paper investigates the combined effects of two distinctive power-type nonlinear terms (with parameters $p,q>1$) in the lifespan of small solutions to semi-linear wave equations. We determine the full region of $(p,q)$ to admit global existence of small solutions, at least for spatial dimensions $n=2, 3$. Moreover, for many $(p,q)$ when there is no global existence, we obtain sharp lower bound of the lifespan, which is of the same order as the upper bound of the lifespan.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1407.6750

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Sep 22, 2013
09/13

by
Kunio Hidano; Chengbo Wang; Kazuyoshi Yokoyama

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We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time $L^2$ estimate for the variable coefficient wave equation plays a central role for this purpose. Assuming radial symmetry, we establish the almost global existence of a strong solution for every small initial data in $H^2 \times H^1$. We also show that the initial value problem is locally well-posed.

Source: http://arxiv.org/abs/1004.3349v2

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41

Sep 22, 2013
09/13

by
Kunio Hidano; Jason Metcalfe; Hart F. Smith; Christopher D. Sogge; Yi Zhou

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The purpose of this paper is to show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small amplitude nonlinear wave equations with power nonlinearities. To achieve this goal, at least for spatial dimensions $n=3$ and 4, we shall show how the aforementioned linear decay estimates can be combined with "abstract Strichartz" estimates for the free...

Source: http://arxiv.org/abs/0805.1673v2