Some remarks on Strichartz estimates for homogeneous wave equation

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.Comment: 28 page

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“…Set δ and δ as in (2). Then there is a positive constant C 2 , such that the following estimates hold …”

Section: Proposition 42mentioning

confidence: 99%

“…Instead, we want to present a proof based on the recent generalized Strichartz estimates of Smith, Sogge and Wang [19] (with the previous radial estimates in Fang and Wang [2]). Lemma 6.1 (Generalized Strichartz estimates).…”

Section: Glassey Conjecture When N = 2 P > P Cmentioning

confidence: 99%

The Glassey conjecture with radially symmetric data

Hidano

Wang

Yokoyama

2012

Journal de Mathématiques Pures et Appliquées

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“…We mention that the related estimates for (1.7) where L 2 θ is replaced by L r θ (with norms of different regularity in the right) were proved by Sterbenz [14] for n ≥ 4 and the authors [4] for the general case n ≥ 2. In the radial case, the estimates (1.7) and the higher dimensional version were proved by the authors in [2] (with previous works in Sogge [10] for n = 3, Sterbenz [14] for n ≥ 3).…”

Section: Introductionmentioning

confidence: 73%

“…When 4 < q < ∞, the Strichartz estimates (1.7) is weaker than the standard Strichartz estimates (see Theorem 3 of [2])…”

Section: Introductionmentioning

confidence: 94%

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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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Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

Hidano

Wang

2019

Sel. Math. New Ser.

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We show new results of wellposedness for the Cauchy problem for the half wave equation with power-type nonlinear terms. For the purpose, we propose two approaches on the basis of the contraction-mapping argument. One of them relies upon the L q t L ∞ x Strichartz-type estimate together with the chain rule of fairly general fractional orders. This chain rule has a significance of its own. Furthermore, in addition to the weighted fractional chain rule established in Hidano, Jiang, Lee, and Wang (arXiv:1605.06748v1 [math.AP]), the other approach uses weighted space-time L 2 estimates for the inhomogeneous equation which are recovered from those for the second-order wave equation. In particular, by the latter approach we settle the problem left open in Bellazzini, Georgiev, and Visciglia (arXiv:1611.04823v1 [math.AP]) concerning the local wellposedness in H s rad (R n ) with s > 1/2.

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Some remarks on Strichartz estimates for homogeneous wave equation

Cited by 41 publications

References 12 publications

The Glassey conjecture with radially symmetric data

The Glassey conjecture with radially symmetric data

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

Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

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