On Blowup in Nonlinear Heat Equations

Blowup behaviour for the nonlinear Klein--Gordon equation

We analyze the blowup behaviour of solutions to the focusing nonlinear Klein--Gordon equation in spatial dimensions $d\geq 2$. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in the conformal and sub-conformal cases. The argument relies on Lyapunov functionals derived from the dilation identity. We also prove that the critical Sobolev norm diverges near the blowup time.

Blowup rate for mass critical rotational nonlinear Schrödinger equations

We consider the blowup rate for blowup solutions to $L^2$-critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the "$\log$-$\log$ law" when the initial data is slightly above the ground state. We also construct minimal mass blowup solutions near the ground state level with distinct blowup rates.

Blowup solutions of Grushin's operator

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.

Blowup for the nonlinear Schrödinger equation with an inhomogeneous damping term in the $L^2$ critical case

We consider the nonlinear Schrödinger equation with $L^2$-critical exponent and an inhomogeneous damping term. By using the tools developed by Merle and Raphael, we prove the existence of blowup phenomena in the energy space $H^1(\mathbb{R})$.

Blowup for the Damped $L^{2}$-Critical Nonlinear Schrödinger Equation

We consider the Cauchy problem for the $L^{2}$-critical damped nonlinear Schrödinger equation. We prove existence and stability of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.

On Blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation

In this paper we study dynamical properties of blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the $L^2$-concentration and the limiting profile with minimal mass of blowup solutions.

A type II blowup for the six dimensional energy critical heat equation

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…

Modulation theory for the flat blowup solutions of nonlinear heat equation

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…

Universal Upper Bound on The Blowup Rate of Nonlinear Schrödinger Equation with Rotation

In this paper, we prove a universal upper bound on the blowup rate of a focusing nonlinear Schrödinger equation with an angular momentum under a trapping harmonic potential, assuming that the initial data is radially symmetric in the weighted Sobolev space. The nonlinearity is in the mass supercritical and energy subcritical regime. Numerical simulations are also presented.

Finite-time blowup for some nonlinear complex Ginzburg-Landau equations

In this article, we review finite-time blowup criteria for the family of complex Ginzburg-Landau equations $u_t = e^{ iθ} [Δu + |u|^αu] + γu$ on ${\mathbb R}^N $, where $0 \le θ\le \frac {π} {2}$, $α>0$ and $γ\in {\mathbb R} $. We study in particular the effect of the parameters $θ$ and $γ$, and the dependence of the blowup time on these parameters.

具有抗干扰爆破剖面的热方程爆破点的反馈近似可控性

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。

Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation

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…

Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow

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.

On the blowup for the $L^2$-critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class

We consider the focusing mass-critical nonlinear Schrödinger equation and prove that blowup solutions to this equation with initial data in $H^s(\R^d)$, $s > s_0(d)$ and $d\geq 3$, concentrate at least the mass of the ground state at the blowup time. This extends recent work by J. Colliander, S. Raynor, C. Sulem, and J. D. Wright, \cite{crsw}, T. Hmidi and S. Keraani, \cite{hk}, and N. Tzirakis, \cite{tzirakis}, on the blowup of the two-dimensional and one-dimensional mass-critical focusing NLS below the energy space to all dimensions $d\ge 3$.

Construction of $L^2$ log-log blowup solutions for the mass critical nonlinear Schrödinger equation

In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schrödinger equation on $\mathbb{R}^{2}$ under rough but structured random perturbations at $L^{2}(\mathbb{R}^2)$ regularity. In particular, by employing probabilistic methods, we provide a construction of a family of $L^{2}(\mathbb{R}^2)$ regularity solutions which do not lie in any $H^{s}(\mathbb{R}^2)$ for any $s>0$, and which blowup according to the log-log dynamics.

On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation

In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{γ_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{γ_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{γ_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{…

基于 $L^2$ 的半线性热方程爆破稳定性与对数修正

$L^2$-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。

Instantaneous blowup and singular potentials on Heisenberg groups

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.

Construction of a stable blowup solution with a prescribed behavior for a non-scaling invariant semilinear heat equation

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…

Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms

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.

色散管理非线性薛定谔方程的适定性与爆破解

Well-posedness and blowup for the dispersion-managed nonlinear Schrödinger equation

本文研究周期色散管理非线性薛定谔方程，建立线性方程的全局Strichartz估计，由此得到三维三次方程的小数据散射结果，并利用virial论证证明爆破解存在。

关于带有空间非均匀非线性项的半线性热方程的稳定爆破剖面

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。

Profile decompositions and Blowup phenomena of mass critical fractional Schrödinger equations

We study, under the radial symmetry assumption, the solutions to the fractional Schrödinger equations of critical nonlinearity in $\mathbb R^{1+d}, d \geq 2$, with Lévy index ${2d}/({2d-1}) < \al < 2$. We firstly prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with mass-critical Hartree type nonlineartity.

Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

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…

Refined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation

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…

到受扰动球面的调和映射热流的稳定爆破

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。

Feedback controllability for blowup points of heat equation

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…

Sign-changing stationary solutions and blowup for the nonlinear heat equation in dimension two

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…

Blowup issues for a class of nonlinear dispersive wave equations

In this paper we consider the nonlinear dispersive wave equation on the real line, $u_t-u_{txx}+[f(u)]_x-[f(u)]_{xxx}+\bigl[g(u)+\frac{f''(u)}{2}u_x^2\bigr]_x=0$, that for appropriate choices of the functions $f$ and $g$ includes well known models, such as Dai's equation for the study of vibrations inside elastic rods or the Camassa--Holm equation modelling water wave propagation in shallow water. We establish a local-in-space blowup criterion (i.e., a criterion involving only the properties of the data $u_0$ in a neighbourhood of a single point) simplifying and extending earlier blowup criter…

Blowup solutions for the nonlinear Schrödinger equation with complex coefficient

We construct a finite time blow up solution for the nonlinear Schrödinger equation with the power nonlinearity whose coefficient is complex number. We generalize the range of both the power and the complex coefficient for the result of Cazenave, Martel and Zhao \cite{CMZ}. As a bonus, we may consider the space dimension $5$. We show a sequence of solutions closes to the blow up profile which is a blow up solution of ODE. We apply the Aubin-Lions lemma for the compactness argument for its convergence.

Global existence versus finite time blowup dichotomy for the system of nonlinear Schrödinger equations

We construct an extremizer for the kinetic energy inequality (except the endpoint cases) developing the concentration-compactness technique for operator valued inequality in the formulation of the profile decomposition. Moreover, we investigate the properties of the extremizer, such as the system of Euler-Lagrange equations, regularity and summability. As an application, we study a dynamical consequence of a system of nonlinear Schrödinger equations with focusing cubic nonlinearities in three dimension when each wave function is restricted to be orthogonal. Using the critical element of the ki…

Quenching, global existence and blowup phenomena in heat transfer

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.

Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2

In this paper we prove finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. This is done in case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned lack of non-decay assumption is essential with respect to keeping the dichotomy finite-time blowup against boundedness of solutions. Namely, we prove that without the non-decay assumption possible asymptotic behaviour…

Profile for the imaginary part of a blowup solution for a complex-valued seminar heat equation

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}{…

Blowup and Scattering problems for the Nonlinear Schrödinger equations

We consider $L^{2}$-supercritical and $H^{1}$-subcritical focusing nonlinear Schrödinger equations. We introduce a subset $PW$ of $H^{1}(\mathbb{R}^{d})$ for $d\ge 1$, and investigate behavior of the solutions with initial data in this set. For this end, we divide $PW$ into two disjoint components $PW_{+}$ and $PW_{-}$. Then, it turns out that any solution starting from a datum in $PW_{+}$ behaves asymptotically free, and solution starting from a datum in $PW_{-}$ blows up or grows up, from which we find that the ground state has two unstable directions. We also investigate some properties of …

On blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation

We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation \[ i\partial_t u -(-Δ)^s u = -|u|^αu, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where $d \geq 2, \frac{d}{2d-1} \leq s <1$, $\frac{4s}{d}<α<\frac{4s}{d-2s}$ and $u_0 \in \dot{H}^{s_{\text{c}}} \cap \dot{H}^s$ is radial with the critical Sobolev exponent $s_{\text{c}}$. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in $\dot{H}^{s_{\text{c}}} \cap \dot{H}^s$. …

Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions

In this paper we consider quasilinear Keller-Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite time of solutions) finite-time blowup result. In the case of a cross-diffusion system we give results which are optimal provided one assumes some proper non-decay of a nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlin…

R^{4+1}中散焦七次复值非线性波动方程的爆破

Blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$

本文通过构造相对论Euler方程的高阶自相似内爆剖面，将R^{4+1}中散焦复值非线性波动方程爆破所需的非线性阶数从29降至7。

On the lifespan of and the blowup mechanism for smooth solutions to a class of 2-D nonlinear wave equations with small initial data

This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu)$ $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and $(c_1'(0))^2+(c_2'(0))^2\neq 0$. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $(u(0,x), \p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$ with $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$, and $\ve>0$ is small, we will…

Ground state mass concentration in the L^2-critical nonlinear Schrodinger equation below H^1

We consider finite time blowup solutions of the $L^2$-critical cubic focusing nonlinear Schrödinger equation on $\R^2$. Such functions, when in $H^1$, are known to concentrate a fixed $L^2$-mass (the mass of the ground state) at the point of blowup. Blowup solutions from initial data that is only in $L^2$ are known to concentrate at least a small amount of mass. In this paper we consider the intermediate case of blowup solutions from initial data in $H^s$, with $1 > s > s_Q$, where $s_Q \le \sQ$. Our main result is that such solutions, when radially symmetric, concentrate at least the mass of …

Global existence and finite time blow-up for the heat flow of H-system with constant mean curvature

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. …

Stable type II blowup for the 7 dimensional 1-corotational energy supercritical harmonic map heat flow

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…

A Fujita-type blowup result and low energy scattering for a nonlinear Schrö\-din\-ger equation

In this paper we consider the nonlinear Schrö\-din\-ger equation $i u_t +Δu +κ|u|^αu=0$. We prove that if $α<\frac {2} {N}$ and $\Im κ<0$, then every nontrivial $H^1$-solution blows up in finite or infinite time. In the case $α>\frac {2} {N}$ and $κ\in {\mathbb C}$, we improve the existing low energy scattering results in dimensions $N\ge 7$. More precisely, we prove that if $ \frac {8} {N + \sqrt{ N^2 +16N }} < α\le \frac {4} {N} $, then small data give rise to global, scattering solutions in $H^1$.

Prevention of blowup via Neumann heat kernel

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…

Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential

We consider a class of the focusing nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu -c|x|^{-2}u = - |u|^αu, \quad u(0)=u_0 \in H^1, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $\frac{4}{d}\leq α\leq \frac{4}{d-2}$ and $c\ne 0$ satisfies $c>-λ(d):=-\left(\frac{d-2}{2}\right)^2$. In the mass-critical case $α=\frac{4}{d}$, we prove the global existence and blowup below ground states for the equation with $d\geq 3$ and $c>-λ(d)$. In the mass and energy intercritical case $\frac{4}{d}<α<\frac{4}{d-2}$, we prove the global existence and blowu…

一类二阶非线性双曲方程的爆破：非线性Jeans不稳定性的简化模型

Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability

本文研究膨胀牛顿宇宙中非线性Jeans不稳定性的二阶非线性双曲方程简化模型，构建了非线性自增爆破解，并估计了增长率以解释宇宙非线性结构观测增长快于经典预测的现象。

On the Global Existence and Blowup Phenomena of Schrödinger Equations with Multiple Nonlinearities

In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \begin{align*} \left\{\begin{array}{ll}&-i u_t=Δu-V(x)u+f(x,|u|^2)u+(W\star|u|^2)u, \quad x\in\mathbb{R}^N, \quad t>0, &u(x,0)=u_0(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*} where $V(x)$ and $W(x)$ are real-valued potentials with $V(x)\geq 0$ and $W$ is even, $f(x,|u|^2)$ is measurable in $x$ and continuous in $|u|^2$, and $u_0(x)$ is a complex-valued function of $x$. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of…

关于超临界波动方程的Strichartz估计与最优爆破稳定性

On Strichartz estimates and optimal blowup stability of supercritical wave equations

本文在相似变量下为径向波动方程建立含空间导数的Strichartz估计，覆盖所有维数及阈值以下正则性，并由此统一推导能量超临界非线性波动方程的最优爆破稳定性，以五次方程为例验证。

Finite-time blowup for a Schrödinger equation with nonlinear source term

We consider the nonlinear Schrödinger equation \[ u_t = i Δu + | u |^αu \quad \mbox{on ${\mathbb R}^N $, $α>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) α\le 4$. Under the additional technical assumptions $α\geq 2$ (and thus $N\leq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${\mathbb R}^N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|^αU$. In the simplest case, $U(t,x)=(|x|^k-α…

关于指数反应扩散方程的精细爆破估计

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。

Finite time blowup and type II rate for harmonic heat flow from Riemannian manifolds

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…

On the blowup and lifespan of smooth solutions to a class of 2-D nonlinear wave equations with small initial data

We are concerned with a class of two-dimensional nonlinear wave equations $\p_t^2u-÷(c^2(u)\na u)=0$ or $\p_t^2u-c(u)÷(c(u)\na u)=0$ with small initial data $(u(0,x),\p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$, where $c(u)$ is a smooth function, $c(0)\not =0$, $x\in\Bbb R^2$, $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$ depend only on $r=\sqrt{x_1^2+x_2^2}$, and $\ve>0$ is sufficiently small. Such equations arise in a pressure-gradient model of fluid dynamics, also in a liquid crystal model or other variational wave equations. When $c'(0)\not= 0$ or $c'(0)=0$, $c"(0)\not= 0$, we establish blowup and d…

Cubic Derivative Nonlinear Schrodinger has Blowup Solutions

This paper is painfully withdrawn.

二维聚焦立方非线性Schrödinger系统极小质量爆破解的分类

Classification of the minimal-mass blowup solutions to the two dimensional focusing cubic nonlinear Schrödinger system

本文研究了质量等于散射阈值的二维聚焦有限和无限耦合立方非线性Schrödinger系统，通过发展多分量调制分析和向量值长时间Strichartz估计，完整分类了有限耦合系统的极小质量爆破解，并证明无限耦合系统解的唯一动力学行为是散射。