Xianfa Song
Basing on the relations between a system of ODE and a system of parabolic equations, we establish some general theories in heat transfer about quenching, global existence and blowup phenomena, obtain the conditions(even watershed) on f(u,v), g(u,v), a(x) and b(x) which let the solution be global existence, quench or blow up, and estimate the bounds for blowup time and quenching time.
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In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar…
This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies the Serrin's condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanish or…
We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear …
We establish the asymptotics of blowup for nonlinear heat equations with superlinear power nonlinearities in arbitrary dimensions and we estimate the remainders.
Blowup analysis for solutions of a general evolution equation with nonlocal diffusion and localized source is performed. By comparison with recent results on global-in-time solutions, a dichotomy result is obtained.
This note is devoted to a simple proof of blowup of solutions for a nonlinear heat equation. The criterion for a blowup is expressed in terms of a Morrey space norm and is in a sense complementary to conditions guaranteeing the global in time existence of solutions. The method goes back to H. Fujita and extends to other nonlinear parabolic equations.
Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by…
Feedback approximate controllability of blowup points for the heat equation with anti-interference blowup profile
本文研究热方程爆破点的反馈近似可控性，证明系统在反馈控制下近似可控且爆破剖面对初始数据稳定，仅发生微小时空平移，表明反馈策略具有抗干扰性。
In this paper we study the quenching problem in nonlinear heat equations with power nonlinearities. For nonlinearities of power p<0 and for an open set of slowly varying initial conditions we prove that the solutions will collapse in a finite time. We find the collapse profile and estimate the remainder.
This paper is concerned with blow-up solutions of the five dimensional energy critical heat equation $u_t=Δu+|u|^\frac{4}{3}u$. A goal of this paper is to show the existence of type II blowup solutions which behave as $\|u(t)\|_\infty\sim(T-t)^{-3k}$ ($k=2,3,\cdots$). Our solutions are the same one formally derived by Filippas, Herrero and Velázquez \cite{Filippas}.
We consider the semilinear heat equation $u_t=Δu+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction property. We show the existence of a new type of blowup solutions by using this property. In fact, we obtain such blowup solutions by connecting a specific blowup solution of $u_t=Δu+|u|^{p-1}u$ and a specific solution of $u_t=Δu-|u|^{q-1}u$, and by adding correction terms.
Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be type II in critical case. When p>p_S=(N+2)/(N-2) for N>=3, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than N-2-4/(p-1), which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collaps…
Stability of nondegenerate ODE type blowup for the Fujita type heat equation
本文研究Fujita型热方程的爆破解渐近行为，将已有非退化ODE型爆破稳定性的结论推广到更一般的情形。
We consider the energy critical semilinear heat equation $$ \left\{\begin{aligned} &\partial_t u-Δu =|u|^{\frac{4}{n-2}}u &\mbox{in } {\mathbb R}^n\times(0,T),\\ &u(x,0)=u_0(x), \end{aligned}\right. $$ where $ n\geq 3$, $u_0\in L^\infty({\mathbb R}^n)$, and $T\in {\mathbb R}^+$ is the first blow up time. We prove that if $ n \geq 7$ and $ u_0 \geq 0$, then any blowup must be of Type I, i.e., \[\|u(\cdot, t)\|_{L^\infty({\mathbb R}^n)}\leq C(T-t)^{-\frac{1}{p-1}}.\] A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysi…
Stable blowup for the harmonic map heat flow into perturbed spheres
该研究针对3至6维、靶流形为单位球面微小扰动的调和映射热流，构造了稳定自相似爆破解并证明其渐近非线性稳定性，采用类似波映射方程相关问题的摄动方法完成论证。
This paper concerns a controllability problem for blowup points on heat equation. It can be described as follows: In the absence of control, the solution to the linear heat system globally exists in a bounded domain $Ω$. While, for a given time $T>0$ and a point $a$ in this domain, we find a feedback control, which is acted on an internal subset $ω$ of this domain, such that the corresponding solution to this system blows up at time $T$ and holds unique point $a$. We show that $a\in ω$ can be the unique blowup point of the corresponding solution with a certain feedback control, and for any fee…
We prove the existence of a (spectrally) stable self-similar blow-up solution $f_0$ to the heat flow for corotational harmonic maps from $\mathbb R^3$ to the three-sphere. In particular, our result verifies the spectral gap conjecture stated by one of the authors and lays the groundwork for the proof of the nonlinear stability of $f_0$. At the heart of our analysis lies a new existence result of a monotone self-similar solution $f_0$. Although solutions of this kind have already been constructed before, our approach reveals substantial quantitative properties of $f_0$, leading to the stability…
In this note, we consider the blowup phenomenon of Grushin's operator. By using the knowledge of probability, we first get expression of heat kernel of Grushin's operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solutions will blow up in finite time.
We consider the energy-supercritical harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, under an additional assumption of 1-corotational symmetry. We are interested by the 7 dimensional case which is the borderline between the Type I blowup regime. We construct for this problem a stable finite time blowup solution under the condition of corotational symmetry that blows up via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{λ(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the rate $$λ(t) \sim \frac{\sqrt{(T-t)}}{|\l…
We study blowup solutions of the 6D energy critical heat equation $u_t=Δu+|u|^{p-1}u$ in $\R^n\times(0,T)$. A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas, Herrero and Velázquez \cite{FilippasHV}. The dimension six is a border case whether a type II blowup can occur or not. Therefore the behavior of the solution is quite different from other cases. In fact, our solution behaves like \[ u(x,t)\approx \begin{cases} λ(t)^{-2}{\sf Q}(λ(t)^{-1}x) & \text{in the inner region: } |x|\simλ(t), -(p-1)^\frac{1}{p-1}(T-t)^{-\frac{1}{p-1}} & \text{in the sel…
In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a at blowup profile, already proved by Bricmont and Kupainen together with Herrero and Velázquez. Though our approach relies on the well celebrated method, based on the reduction of the problem to a finite dimensional one, then the use of a topological shooting method to solve the latter, the novelty of our approach lays in the use of a modulation technique to control the projection of the zero eigenmode arising in the problem. Up to our knowledge, this is the first ti…
In this paper, we consider the standard semilinear heat equation \begin{eqnarray*} \partial_t u = Δu + |u|^{p-1}u, \quad p >1. \end{eqnarray*} The determination of the (believed to be) generic blowup profile is well-established in the literature, with the solution blowing up only at one point. Though the blow-up of the gradient of the solution is a direct consequence of the single-point blow-up property and the mean value theorem, there is no determination of the final blowup profile for the gradient in the literature, up to our knowledge. In this paper, we refine the construction technique of…
In this paper, we study the formation of finite time singularities for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by constructing a Lyapunov functional. Moreover, it is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.
$L^2$-based stability of blowup with log correction for semilinear heat equation
本文提出了一种基于 $L^2$ 的半线性热方程爆破稳定性分析方法，通过稳定归一化和加权 $H^k$ 稳定性，避免了拓扑论证和谱分析，适用于数值轮廓情况，并扩展了稳定性框架的应用范围。
In this paper we generalize the instantaneous blowup result from [3] and [15] to the heat equation perturbed by singular potentials on the Heisenberg group.
We consider the semilinear heat equation \begin{eqnarray*} \partial_t u = Δu + |u|^{p-1} u \ln ^α( u^2 +2), \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $p > 1$ and $ α\in \mathbb{R}$. Unlike the standard case $α= 0$, this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time $T$ only at one blowup point $a$, according to the following asymptotic dynamics: \begin{eqnarray*} u(x,t) \sim ψ(t) \left(1 + \frac{(p-1)|x-a|^2}{4p(T -t)|\ln(T -t)|} \right)^{-\frac{1}{p-1}} \text{ as } t \to T, \end{eqnarray*} where $ψ(t)$ is the unique p…
We consider the Cauchy problem of the nonlinear heat equation $u_t -Δu= u^{b},\ u(0,x)=u_0$, with $b\geq 2$ and $b\in \mathbb{N}$. We prove that initial data $u_0\in \mathcal{S}(\mathbb{R}^{n})$ (the Schwartz class)arbitrarily small in the scale invariant Besov-norm$\dot B^{-2/b}_{n(b-1) b/2,q}(\mathbb{R}^{n})$, can produce solutions that blow up in finite time. The case $b=3$ answers a question raised by Yves Meyer.Our result also proves that the smallness assumption put in an earlier work by C. Miao, B.~Yuan and B. Zhang, for the global-in-time solvability, is essentially optimal.
Stable blowup profile for a semilinear heat equation with spatially inhomogeneous nonlinearity
研究了带有额外的发散Hénon型非线性的聚焦半线性热方程，分析了其在小非径向扰动下的稳定性，并证明了在c接近c*时的稳定爆破。
We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\partial_t U = ΔU + α|\nabla U|^2 + e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad α> -1.$$ We construct for this equation a solution which blows up in finite time $T > 0$ and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time $T$ simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some senses, resulting in the c…
Consider the nonlinear heat equation v_t-Δv=|v|^{p-1}v in the unit ball of R^2, with Dirichlet boundary condition. Let u_{p,K} be a radially symmetric, sign-changing stationary solution having a fixed number K of nodal regions. We prove that the solution of the equation with initial value λu_{p,K} blows up in finite time if |λ-1|>0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of $u_{p,K}$ and of the linearized operator L= -Δ- p |u_{p,K}|^{p-1}. To show this we consider the linearized operator L= -Δ- p|u_p|^{p-1} and study t…
Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows
本文研究了 $d \ge 11$ 时高维Yang-Mills热流的奇异性形成，通过构造含时特征向量并结合最大值原理与半群逐点估计，构造了非自相似爆破解并证明了其渐近稳定性。
In this paper, we consider the following complex-valued semilinear heat equation \begin{eqnarray*} \partial_t u = Δu + u^p, u \in \mathbb{C}, \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $ p \in \mathbb{N}, p \geq 2$. We aim at constructing for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a$, with the following estimates for the final profile \begin{eqnarray*} u(x,T) &\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}}, u_2(x,T) &\sim & \frac{2 p}{(p-1)^2} \left[ \frac{ (p-1)^2|x-a|^2}{…
In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on $(0,T) \times Ω:$ \begin{eqnarray*} \partial_t u = Δu +\displaystyle \frac{λ}{ (1-u)^2 \left( 1 +\displaystyle γ\int_Ω \frac{1}{1-u} dx \right)^2}, \quad 0 \leq u <1, \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^n$ and $λ, γ> 0$. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point $a \in Ω$. In particular, we give a description of the quenching behavior according to the following final profile $…
In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H-system in a bounded smooth domain of $R^2$. Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the $L^2$ norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. …
Consider the heat equation $u_t-Δu=0$ on a bounded $C^2$ domain $Ω$ in $\mathbb{R}^{n}(n\geq 2)$ with any positive initial data. If a superlinear radiation law $\frac{\partial u}{\partial n}=u^{q}$ with $q>1$ is imposed on a partial boundary $Γ_1\subseteq\partialΩ$ which has a positive surface area, then it has been known that the solution $u$ blows up in finite time. However, if the partial boundary, on which the superlinear radiation law is prescribed, is shrinking and is denoted as $Γ_{1,t}$ at time $t$, then the solution may exist globally as long as the surface area $|Γ_{1,t}|$ of $Γ_{1,t…
On refined blowup estimates for the exponential reaction-diffusion equation
本文研究了具有指数非线性的半线性热方程的径向递减解，给出了最终爆破轮廓和精细时空行为的精确上界估计的简单证明，并建立了一个包含前述特例的全局时空上界估计。
In this paper, we will study the existence of finite time singularity to harmonic heat flow and their formation patterns. After works of Coron-Ghidaglia, Ding and Chen-Ding, one knows blow-up solutions under smallness of initial energy for m>=3. soon later, 2 dimensional blowup solutions were found by Chang-Ding-Ye. The first part of this paper is devoted to construction of new examples of finite time blow-up solutions without smallness conditions for 3<=m<7. In fact, when considering rotational symmetric harmonic heat flow from B_1\subset R^m to S^m\subset R^{m+1}, we will prove that the maxi…
This paper is concerned with the blowup phenomenon of stochastic parabolic equations both on bounded domain and in the whole space. We introduce a new method to study the blowup phenomenon on bounded domain. Comparing with the existing results, we delete the assumption that the solutions to stochastic heat equations are non-negative. Then the blowup phenomenon in the whole space is obtained by using the properties of heat kernel. We obtain that the solutions will blow up in finite time for nontrivial initial data.
We study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see definition in the paper). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution h…
In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = Δu +a |u|^αu, \; t\in(0,T),\; x=(x_1,\,\cdots,\, x_N)\in {\mathbb R}^N,\; a = \pm 1,\; α>0;$ with initial value $u(0)\in L^1_{\rm{loc}}\left({\mathbb R}^N\setminus\{0\}\right)$, anti-symmetric with respect to $x_1,\; x_2,\; \cdots,\; x_m$ and $|u(0)|\leq C(-1)^m\partial_{1}\partial_{2}\cdot \cdot \cdot \partial_{m}(|x|^{-γ})$ for $x_1>0,\; \cdots,\; x_m>0,$ where $C>0$ is a constant, $m\in \{1,\; 2,\; \cdots,\; N\},$ $0<γ<N$ and $0<α<2/(γ+m).$ This gives a local existence result with highly singular initi…
In this paper, we consider the following semi-linear complex heat equation \begin{eqnarray*} \partial_t u = Δu + u^p, u \in \mathbb{C} \end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be non integer and even irrational. We construct for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a.$ Moreover, we also describe the asymptotics of the solution by the following final profiles: \begin{eqnarray*} u(x,T) &\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}}…
We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\partial_t u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{λ(t)}\right),$$ where $Q$ is the stationary solution of the equation and …
This paper considers the immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations. The viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., $μ=\barμ θ^α, ~ λ=\barλ θ^α,~ κ=\barκ θ^α$, for constants $0\leq α\leq 1/(γ-1)$, $\barμ>0,~2\barμ+n\barλ\geq0,~\barκ\geq 0$, and adiabatic exponent $γ>1$). With prescribed decaying rate of the initial density across the vacuum boundary, we prove that: (1…
It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomenon, it remain to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also give an affirmative answer t…
On the semilinear heat equation with the Grushin operator
本文研究Grushin算子的热方程，给出热核表达式并证明其衰减性与恒等逼近性质，进而估计热半群，并证明非线性Cauchy问题温和解的存在唯一性、连续依赖及爆破替代，还给出全局存在性结果。
We analyse the behaviour of solutions of the linear heat equation in ${\mathbb R}^d$ for initial data in the classes $M_\varepsilon({\mathbb R}^d)$ of Radon measures with $\int_{{\mathbb R}^d}{\rm e}^{-\varepsilon|x|^2}\,{\rm d}|u_0|<\infty$. We show that these classes are in some sense optimal for local and global existence of non-negative solutions: in particular $M_0({\mathbb R}^d)=\cap_{\varepsilon>0}M_\varepsilon({\mathbb R}^d)$ consists precisely of those initial data for which the a solution of the heat equation can be given for all time using the heat kernel representation formula. Aft…
We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of the Weinkove solution under small equivariant perturbations and thus extend a result by the second author and Donninger for $d=5$ to higher dimensions. At the same time, we provide a general framework for proving stability of self-similar blowup solutions to a large clas…
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.
Co-existence of Type II blow-ups with multiple blow-up rates for five-dimensional heat equation with critical nonlinear boundary conditions
该研究针对带临界非线性边界条件的五维热方程，构造了在不同边界点处具有不同II型爆破速率的有限时间爆破解，首次证明了多重速率II型爆破的共存性，并发展了适配该类不稳定爆破的统一线性理论。
We study $O(d)$-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flow from $B^4(0, 1)$ into $S^4$. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible $O(4)$-equivariant biharmonic maps from $\mathbf{R}^4$ into $S^4$, and we show that there exists, in contrast to…
Non-blowup of stochastic heat equations by noise
本文证明在热方程中加入乘性噪声（布朗运动驱动）可阻止由超线性增长系数导致的有限时间爆破，在特定增长条件下随机热方程存在全局解，截断与比较技术是关键。
Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions
本文研究了高维调和映射热流的奇异性形成，通过构造旋转对称目标流形，证明了稳定共旋自相似收缩解的存在性，揭示了 Cauchy 问题的稳定爆破机制。
Asymptotic blow-up behavior for the semilinear heat equation with non scale invariant nonlinearity
本文研究了非尺度不变非线性半线性热方程的渐近爆破行为，证明了一大类非线性项的正解在爆破点附近趋于对应 ODE 解，并给出了爆破集的紧致性等性质。
Quantized slow blowup dynamics for the energy-critical Schrödinger map flow
该研究针对同伦度为1的等变初始数据，构造了接近基态调和映射的初始数据，证明能量临界薛定谔映射流会以量化速率有限时间爆破，引入了新的李雅普诺夫泛函完成证明。
Instantaneous everywhere-blowup of parabolic SPDEs
本文证明随机热方程在Osgood条件下解几乎必然瞬时处处爆破，方法结合微分不等式击中时间界与Malliavin微积分分析随机卷积空间增长。
This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small parameter $\ep$ when $p_F(N)\leq p\leq p_0(N+μ)$, where $p_F(N)$ denotes the Fujita exponent for the nonlinear heat equations and $p_0(n)$ denotes the Strauss exponent for nonlinear wave equation in $n$-dimension with $μ=0$. Consequently, by connecting the result of D'Abbicco--Lucente--Reissig 2015, our result clarifies the threshold exponent $p_0(N+μ)$ for di…
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