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Ziheng Tu

Publications and source records attributed to Ziheng Tu.

10 recordsLinked to original sources
arXiv2021

Small data blow-up for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds

In the present paper, we study the Cauchy problem for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds. Our aim of this paper is to prove a small data blow-up result and an upper estimate of lifespan of the problem for a suitable compactly supported initial data in the subcritical and critical cases of the Strauss type. The proof is based on the framework of the argument in the paper [17]. One of our new contributions is to construct two families of special solutions to the free equation (see (2.16) or (2.18) as the test functions and prove their several properties. We emphasize that the system with two different propagation speeds is treated in this paper and the assumption on the initial data is improved from the point-wise positivity to the integral positivity.

math.AP↗
arXiv2021

Lifespan estimates for semilinear wave equations with space dependent damping and potential

In this work, we investigate the influence of general damping and potential terms on the blow-up and lifespan estimates for energy solutions to power-type semilinear wave equations. The space-dependent damping and potential functions are assumed to be critical or short range, spherically symmetric perturbation. The blow up results and the upper bound of lifespan estimates are obtained by the so-called test function method. The key ingredient is to construct special positive solutions to the linear dual problem with the desired asymptotic behavior, which is reduced, in turn, to constructing solutions to certain elliptic "eigenvalue" problems.

math.AP↗
arXiv2021

Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass

In the present paper, we study small data blow-up of the semi-linear wave equation with a scattering dissipation term and a time-dependent mass term from the aspect of wave-like behavior. The Strauss type critical exponent is determined and blow-up results are obtained to both sub-critical and critical cases with corresponding upper bound lifespan estimates. For the sub-critical case, our argument does not rely on the sign condition of dissipation and mass, which gives the extension of the result in \cite{Lai-Sch-Taka18}. Moreover, we show the blow-up result for the critical case which is a new result.

math.AP↗
arXiv2019

A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied.

math.AP↗
arXiv2019

Strauss exponent for semilinear wave equations with scattering space dependent damping

It is believed or conjectured that the semilinear wave equations with scattering space dependent damping admit the Strauss critical exponent, see Ikehata-Todorova-Yordanov \cite{ITY}(the bottom in page 2) and Nishihara-Sobajima-Wakasugi \cite{N2}(conjecture iii in page 4). In this work, we are devoted to showing the conjecture is true at least when the decay rate of the space dependent variable coefficients before the damping is larger than 2. Also, if the nonlinear term depends only on the derivative of the solution, we may prove the upper bound of the lifespan is the same as that of the solution of the corresponding problem without damping. This shows in another way the \lq\lq hyperbolicity" of the equation.

math.AP↗
arXiv2019

Lifespan of semilinear generalized Tricomi equation with Strauss type exponent

In this paper, we consider the blow-up problem of semilinear generalized Tricomi equation. Two blow-up results with lifespan upper bound are obtained under subcritical and critical Strauss type exponent. In the subcritical case, the proof is based on the test function method and the iteration argument. In the critical case, an iteration procedure with the slicing method is employed. This approach has been successfully applied to the critical case of semilinear wave equation with perturbed Laplacian or the damped wave equation of scattering damping case. The present work gives its application to the generalized Tricomi equation.

math.AP↗
arXiv2018

Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity

In this paper, we study the blow-up of solutions for semilinear wave equations with scale-invariant dissipation and mass in the case in which the model is somehow 'wave-like'. A Strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems. Two blow-up results are obtained for the sub-critical case and for the critical case, respectively. In both cases, an upper bound lifespan estimate is given.

math.AP↗
arXiv2017

Life-Span of Semilinear Wave Equations with Scale-invariant Damping: Critical Strauss Exponent Case

The blow up problem of the semilinear scale-invariant damping wave equation with critical Strauss type exponent is investigated. The life span is shown to be: $T(\varepsilon)\leq C\exp(\varepsilon^{-2p(p-1)})$ when $p=p_S(n+μ)$ for $0<μ<\frac{n^2+n+2}{n+2}$. This result completes our previous study \cite{Tu-Lin} on the sub-Strauss type exponent $p<p_S(n+μ)$. Our novelty is to construct the suitable test function from the modified Bessel function. This approach might be also applied to the other type damping wave equations.

math.AP↗
arXiv2016

Well-posedness for the Cauchy problem of spatially weighted dissipative equation

This paper mainly investigates the Cauchy problem of the spatially weighted dissipative equation with initial data in the weighted Lebesgue space. A generalized Hankel Transform is introduced to derive the analytical solution and a special Young's Inequality has been applied to prove the space-time estimates for this type of equation.

math.AP↗