Minimal regularity solutions of semilinear generalized Tricomi equations

Ruan, Zhuoping; Witt, Ingo; Yin, Huicheng

doi:10.2140/pjm.2018.296.181

Cited by 12 publications

(16 citation statements)

References 22 publications

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“…t u − t m ∆u = |u| p , (t, x) ∈ R 1+n + , u(0, x) = u 0 (x), ∂ t u(0, x) = u 1 (x), (1.3) where m ∈ N, p > 1, n ≥ 2. For the local existence and regularity of solution u to (1.3) under weak regularity assumptions on (u 0 , u 1 ), the reader may consult [24,25,26,34,35]. If u i ∈ C ∞ 0 (R n ) (i = 0, 1) Yagdjian in [34] proved the blowup and global existence of u when p belongs to different intervals.…”

Section: Daoyin He Ingo Witt and Huicheng Yinmentioning

confidence: 99%

“…Now let us turn back to the 1-D Cauchy problem for Tricomi equation (1.1). Since the local existence of weak solution u to (1.1) under minimal regularity assumptions has been established in [26] and [34], without loss of generality, as in [13,14], we focus on the global small-data weak solution problem to (1.1) starting at a fixed positive time. Moreover, in order to establish the global existence result, we need to apply the Strichartz estimates with a characteristic weight ((φ(t) + M ) 2 − |x| 2 ) γ , however, the characteristic cone for Tricomi operator is φ 2 (t) = |x| 2 , which admits a cusp singularity at t = 0, due to this difficulty, we can only establish inhomogeneous Strichartz estimates for the case t is away from zero.…”

Section: Daoyin He Ingo Witt and Huicheng Yinmentioning

confidence: 99%

“…Thus, without loss of generality, we make some reduction of the problem (1.1). More specifically, by the local existence theory in [26] and [34], for givenε > 0, there exists a fixed smallT > 0, such that if ε <ε, then the lifespan T of the local solution of (1.1) satisfies T ≥T . To this end, it is reasonable that one replaces nonlinearity |u| p in (1.1) with the nonlinear function…”

Section: Daoyin He Ingo Witt and Huicheng Yinmentioning

confidence: 99%

“…The authors of [11,19,20,21,22] have obtained a series of interesting results on the existence and uniqueness of solutions u to the semilinear Tricomi equation ∂ 2 t u − t∆u = f (t, x, u) in bounded domains, under certain restrictions on the nonlinearity f (t, x, u). The authors of [2,24,25,26] have established the local existence as well as the singularity structure of low regularity solutions to the Cauchy problem for semilinear Tricomi equations in the degenerate hyperbolic region and the elliptic-hyperbolic mixed region, respectively. We have additionally given a complete study of the global existence versus blowup problem for small-data solutions u to the semilinear Tricomi equation ∂ 2 t u − t∆u = |u| p for n ≥ 2 (see [12,13,14]).…”

Section: The Linear Equation ∂mentioning

confidence: 99%

See 3 more Smart Citations

On the strauss index of semilinear tricomi equation

He¹,

Witt²,

Yin³

2020

Communications on Pure &Amp; Applied Analysis

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In our previous papers, we have given a systematic study on the global existence versus blowup problem for the small-data solution u of the multi-dimensional semilinear Tricomi equation ∂ 2 t u − t ∆u = |u| p , u(0, •), ∂tu(0, •) = (u 0 , u 1), where t > 0, x ∈ R n , n ≥ 2, p > 1, and u i ∈ C ∞ 0 (R n) (i = 0, 1). In this article, we deal with the remaining 1-D problem, for which the stationary phase method for multi-dimensional case fails to work and the large time decay rate of u(t, •) L ∞ x (R) is not enough. The main ingredient of the proof in this paper is to use the structure of the linear equation to get the suitable decay rate of u in t, then the crucial weighted Strichartz estimates are established and the global existence of solution u is proved when p > 5.

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Section: Daoyin He Ingo Witt and Huicheng Yinmentioning

confidence: 99%

Section: Daoyin He Ingo Witt and Huicheng Yinmentioning

confidence: 99%

Section: Daoyin He Ingo Witt and Huicheng Yinmentioning

confidence: 99%

Section: The Linear Equation ∂mentioning

confidence: 99%

See 2 more Smart Citations

On the strauss index of semilinear tricomi equation

He¹,

Witt²,

Yin³

2020

Communications on Pure &Amp; Applied Analysis

Self Cite

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“…We now give another example of prior work that is closely connected to our main results. In [87], the authors proved local well-posedness in homogeneous Sobolev spaces on domains of the form [0, T ) × R n for semilinear Tricomi equations of the form…”

Section: Introductionmentioning

confidence: 99%

Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

Speck

2017

Analysis & PDE

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Abstract. In three spatial dimension, we study the Cauchy problem for the wave equation −∂ 2 t Ψ + (1 + Ψ) P ∆Ψ = 0 for P ∈ {1, 2}. We exhibit a form of stable Tricomi-type degeneracy formation that has not previously been studied in more than one spatial dimension. Specifically, using only energy methods and ODE techniques, we exhibit an open set of data such that Ψ is initially near 0 while 1 + Ψ vanishes in finite time. In fact, generic data, when appropriately rescaled, lead to this phenomenon. The solution remains regular in the following sense: there is a high-order L 2 -type energy, featuring degenerate weights only at the top-order, that remains bounded. When P = 1, we show that any C 1 extension of Ψ to the future of a point where 1 + Ψ = 0 must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar of the Lorentzian metric corresponding to the wave equation blows up at those points. Thus, our results show that curvature blowup does not always coincide with singularity formation in the solution variable. Similar phenomena occur when P = 2, where the vanishing of 1 + Ψ corresponds to the failure of strict hyperbolicity, although the equation is hyperbolic at all values of Ψ.The data are compactly supported and are allowed to be large or small as measured by an unweighted Sobolev norm. However, we assume that initially, the spatial derivatives of Ψ are nonlinearly small relative to |∂ t Ψ|, which allows us to treat the equation as a perturbation of the ODE d 2 dt 2 Ψ = 0. We show that for appropriate data, ∂ t Ψ remains quantitatively negative, which simultaneously drives the degeneracy formation and yields a favorable spacetime integral in the energy estimates that is crucial for controlling some top-order error terms. Our result complements those of Alinhac and Lindblad, who showed that if the data are small as measured by a Sobolev norm with radial weights, then the solution is global.

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Sharp lifespan estimates for subcritical generalized semilinear Tricomi equations

Sun

2021

Math Methods in App Sciences

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In this paper, we investigate the Cauchy problem of the subcritical semilinear generalized Tricomi equations with dimension n ≥ 3. Under the assumption that the initial data are smooth and compactly supported, the Cauchy problem is shown to be locally well‐posed and the lower bound estimate of the lifespan is established. We also obtain the upper bound of the lifespan with the same scale; hence, the sharp lifespan is determined. The proof is based on the improved versions of the weighted Strichartz estimates.

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Minimal regularity solutions of semilinear generalized Tricomi equations

Cited by 12 publications

References 22 publications

On the strauss index of semilinear tricomi equation

On the strauss index of semilinear tricomi equation

Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

Sharp lifespan estimates for subcritical generalized semilinear Tricomi equations

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