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Jiayun Lin

Publications and source records attributed to Jiayun Lin.

6 recordsLinked to original sources
arXiv2026

Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent

In this paper, we investigate the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-Δ)^{σ/2} u(t,x)+b(t) \partial_t u(t,x) =f(u(t,x)),\ &(t,x)\ \in [0,T)\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x),\ &x\ \in\ \mathbb{R}^N, \end{cases} $$ where the damping coefficient behaves as $b(t)\approx (1+t)^{-β}$. We first establish local existence and uniqueness of mild and strong solutions in weighted Sobolev spaces by deriving weighted Sobolev estimates for the linear propagators and overcoming the spatial-weight coupling in the nonlinearity through the Bessel potential estimates. Then, in the subcritical and critical cases $1<p\leq p_c:=1+\frac σN$, we derive the upper estimates of the lifespan for the fractional Laplacian with $0<σ<2$ and time-dependent damping $β\in [-1, 1)$ via the framework of ordinary differential inequalities. The blow-up results, together with the global existence in the supercritical case obtained in \cite{LI}, show that the critical exponent for the fractional wave equation is $p_c=1+\fracσ{N}$ for $0<σ<2$. Moreover, combining with the lower estimate of lifespan derived in \cite{LI}, we conclude that the estimate in this paper is sharp. Note that our result for the critical case is completely new even in the classical case $b(t)=1$. We also consider the case of $β=1$, and obtain the upper estimate of the lifespan.

math.AP↗
arXiv2026

Weighted Fractional Commutator Estimate and Application to the Fractional Wave Equations with Time-dependent Damping

In this paper, we first establish a weighted bilinear fractional commutator estimate involving the spatial weight $\langle x \rangle^{2m}$. This estimate provides a powerful tool to resolve the difficulties arising from the nonlocality of the fractional Laplacian $(-Δ)^{σ/2}$ and the interaction between the unbounded weights and the nonlocal operator at infinity. We then apply this result to investigate the Cauchy problem for the fractional wave equation with time-dependent damping and source nonlinearity: $$ \begin{cases} \partial_t^2 u(t,x) + (-Δ)^{σ/2} u(t,x) + b(t) \partial_t u(t,x) = |u(t,x)|^p, & (t,x) \in (0,T) \times \mathbb{R}^N, \\ u(0,x) = \varepsilon u_0(x), \quad \partial_t u(0,x) = \varepsilon u_1(x), & x \in \mathbb{R}^N, \end{cases} $$ where $σ\in (0,2)$, the initial data scale with a small parameter $\varepsilon > 0$, and the time-dependent damping coefficient $b(t)$ satisfies $b(t) \approx (1+t)^{-β}$ for $β\in [-1, 1)$. Utilizing scaling variables and constructing a weighted energy framework based on Theorem 1.1, we derive a priori differential inequalities for the total energy. For the supercritical case $p > p_c := 1 + \fracσ{N}$, we prove the small-data global existence and uniqueness of strong solutions in weighted Sobolev spaces $H^{σ/2,m}(\mathbb{R}^N) \times H^{0,m}(\mathbb{R}^N)$ by closing a bootstrap argument. For the subcritical ($1 < p < p_c$) and critical ($p = p_c$) cases, we establish precise lower bounds for the lifespan $T_\varepsilon$ of blow-up solutions via delicate ODE analysis. Together with the corresponding upper estimates, our results fully determine the sharp lifespan estimates for all $β\in [-1, 1)$, confirming that $p_c = 1 + \fracσ{N}$ is indeed the critical exponent for the fractional wave equation with time-dependent damping.

math.AP↗
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↗
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↗
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↗