For the quasilinear wave equation
\partial_t^2u - Δu = u_t u_{tt},
we analyze the long-time behavior of classical solutions with small (not rotationally invariant) data. We give a complete asymptotic expansion of the lifespan and describe the solution close to the blowup point. It turns out that this solution is a ``blowup solution of cusp type,'' according to the terminology of the author.
We prove that, unlike in several space dimensions, there is no critical (nonlinear) diffusion coefficient for which solutions to the one dimensional quasilinear Smoluchowski-Poisson equation with small mass exist globally while finite time blowup could occur for solutions with large mass.
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…
For the 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(\nabla u)\partial_{ij}u=0$ with coefficients independent of the solution $u$, a blowup result for small data solutions has been established in [1,2] provided that the null condition does not hold and a generic nondegeneracy condition on the initial data is fulfilled. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(u, \nabla u)\partial_{ij}u=0$ with coefficients that depend simultaneously on $u$ and $\nabla u$. When the null condition does not hold an…
We prove a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. The singularity is more severe than a shock in that the solution itself blows up like the log of the distance to the blowup-time. We assume that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the "elliptic part" of the wave operator vanishes precisely at the singular points. The initial data are compactly supported and can be small or la…
Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schrödinger equations
China People’s Police University（中国人民警察大学）；Hebei Key Laboratory of Information Support Technology for Smart Policing, China People’s Police University（中国人民警察大学智慧警务信息支撑技术河北省重点实验室）；Tianjin University（天津大学）；Xinjiang University of Technology（新疆理工学院）
For the first order 1D $n\times n$ quasilinear strictly hyperbolic system $\partial_tu+F(u)\partial_xu=0$ with $u(x, 0)=\varepsilon u_0(x)$, where $\varepsilon>0$ is small, $u_0(x)\not\equiv 0$ and $u_0(x)\in C_0^2(\mathbb R)$, when at least one eigenvalue of $F(u)$ is genuinely nonlinear, it is well-known that on the finite blowup time $T_{\varepsilon}$, the derivatives $\partial_{t,x}u$ blow up while the solution $u$ keeps to be small. For the 1D scalar equation or $2\times 2$ strictly hyperbolic system (corresponding to $n=1, 2$), if the smooth solution $u$ blows up in finite time, then the…
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…
In three spatial dimensions, we study the Cauchy problem for the model wave equation $- \partial_t^2 Ψ+ (1 + Ψ)^P ΔΨ= 0$ for $P \in \lbrace 1,2 \rbrace$. We exhibit a stable form of finite-time Tricomi-type degeneracy formation that has not previously been studied for quasilinear wave equations. Specifically, using only energy methods and ODE-type techniques, we exhibit an open (in an appropriate Sobolev topology) set of data such that $Ψ$ is initially near $0$ while $1 + Ψ$ vanishes in finite time. In fact, generic data profiles, when appropriately rescaled, lead to the vanishing of $1 + Ψ$ i…
We construct examples of finite time singularity from smooth data for linear uniformly parabolic systems in the plane. We obtain similar examples for quasilinear systems with coefficients that depend only on the solution.
This paper is concerned with the small smooth data problem for the 3-D nonlinear wave equation $\partial_t^2u-\left (1+u+\p_t u\right)Δu=0$. This equation is prototypical of the more general equation $\dsize\sum_{i,j=0}^3g_{ij}(u, \nabla u)\partial_{ij}u=0$, where $x_0=t$ and $g_{ij}(u, \nabla u)=c_{ij}+d_{ij}u+\dsize\sum_{k=0}^3e_{ij}^k\partial_ku+O(|u|^2+|\nabla u|^2)$ are smooth functions of their arguments, with $c_{ij}, d_{ij}$ and $e_{ij}^k$ being constants, and $d_{ij}\neq0$ for some $(i,j)$; moreover, $\dsize\sum_{i,j,k=0}^3e_{ij}^k(\partial_ku)\p_{ij} u$ does not fulfill the null cond…
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation \[ u_t-\mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right)=-|u|^{β-1}u+α|u|^{q-2}u, \] where $p>1,β>0$, $q\geq1$ and $α>0$. By using Gagliardo-Nirenberg type inequality, energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.
Blowup of smooth solutions to 2D quasilinear wave equations with higher-order null conditions and short pulse initial data
School of Mathematical Sciences and IMS, Nanjing Normal University（南京师范大学数学科学学院和集成数学科学研究所）；Institute of Mathematical Sciences, The Chinese University of Hong Kong（香港中文大学数学科学研究所）
Unboundedness of solutions is shown to occur in a one-dimensional quasilinear parabolicparabolic chemotaxis system for any initial mass. Our result is also independent of the relation between the speeds of the diffusion of cells and chemoattractant. The proof is achieved by contradiction. It uses a virial type inequality together with a boundedness from below of the Liapunov functional associated to the system. Moreover, the same method is applied also in higher dimensions and an unboundedness result is shown in the case of critical quasilinear fully parabolic Keller-Segel system for mass larg…
In this paper, we are concerned with the 3-D quasilinear wave equation $ \ds\sum_{i,j=0}^3g^{ij}(u, \p u)\p_{ij}^2u$ $=0$ with $(u(0,x), \p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$, where $x_0=t$, $x=(x_1, x_2, x_3)$, $\p=(\p_0, \p_1, ..., \p_3)$, $u_0(x), u_1(x)\in C_0^\infty(\Bbb R^3)$, $\ve>0$ is small enough, and $g^{ij}(u, \p u)=g^{ji}(u, \p u)$ are smooth in their arguments. Without loss of generality, one can write $g^{ij}(u, \p u)=c^{ij}+d^{ij}u+\ds\sum_{k=0}^3e^{ij}_k\p_ku+O(|u|^2+|\p u|^2)$, where $c^{ij}, d^{ij}$ and $e^{ij}_k$ are some constants, and $\ds\sum_{i,j=0}^3c^{ij}\p_{ij}^2=-\s…
We prove a stable shock formation result for a large class of systems of quasilinear wave equations in two spatial dimensions. We give a precise description of the dynamics all the way up to the singularity. Our main theorem applies to systems of two wave equations featuring two distinct wave speeds and various quasilinear and semilinear nonlinearities, while the solutions under study are (non-symmetric) perturbations of simple outgoing plane symmetric waves. The two waves are allowed to interact all the way up to the singularity. Our approach is robust and could be used to prove shock formati…
Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions, provided that the initial data are compactly supported and sufficiently small in Sobolev norm. In this work, Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions. A natural question then arises whether the time-growth is a true phenomena, despite the possible conservation of basic energy. Analogous problems are also of central importance for Schrödinger e…
We carry on our studies related to the fully parabolic quasilinear Keller-Segel system started in [6] and continued in [7]. In the above mentioned papers we proved finite-time blowup of radially symmetric solutions to the quasilinear Keller-Segel system if the nonlinear chemosensitivity is strong enough and an adequate relation between nonlinear diffusion and chemosensitivity holds. On the other hand we proved that once chemosensitivity is weak enough solutions exist globally in time. The present paper is devoted to looking for critical exponents distinguishing between those two behaviors. Mor…
In this paper, we deal with the Cauchy problem of the quasilinear Schödinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=Δu+2uh'(|u|^2)Δh(|u|^2)+(W(x)\ast|u|^2)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $W(x)$ are some real valued functions. Our focus is to investigate how the interplay between the potential $W(x)$ and the quasilinear presence $h(s)$ affects the blowup in finite time and global existence of the solution. In a special, we can obtain the watershed condition on $W(x)$ in the following sens…
We show that the one-dimensional fully parabolic Keller-Segel system with nonlinear diffusion possesses global-in-time solutions, provided the nonlinear diffusion is equal to (1+u)^{-α}, for α< 1, independently on the volume of the initial data. We also show that in the critical case, i.e. for α= 1, the same result holds for initial masses smaller than a prescribed constant. Additionally, we prove existence of initial data for which solution blows up in a finite time for any nonlinear diffusion integrable at infinity. Thus we generalize the known blowup result of parabolic-elliptic case to the…
In this article we develop the local wellposedness theory for quasilinear Maxwell equations in $H^m$ for all $m \geq 3$ on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in $H^3$ is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in $H^m$ with $m \geq 3$ of the corresponding initial boundary value problem if the material laws and the data are accor…
Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime
Georgia Institute of Technology（佐治亚理工学院）；Vanderbilt University（范德堡大学）
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 $d\ge 3$.
We study a class of semilinear elliptic equations with constraints in higher dimension. It is known that several mathematical structures of the problem are closed to those of the Liouville equation in dimension two. In this paper, we establish a classification of entire solutions, the $\sup + \inf$ type inequality and the quantized blowup mechanism.
We consider equivariant wave maps from $\mathbb{R}^{d+1}$ to $\mathbb{S}^d$ in supercritical dimensions $3\leq d\leq 6$. Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable manifold of a self-similar solution with one instability.
This is the content of the talk the author gave in the section of partial differential equations at the International Congress of Chinese Mathematicians, 2013, Taipei.
We consider two-dimensional versions of the Keller--Segel model for the chemotaxis with either classical (Brownian) or fractional (anomalous) diffusion. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Moreover, the impact of the consumption term on the global-in-time existence of solutions is analyzed for the classical Keller--Segel system.
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}.
The large time behaviour of nonnegative solutions to a quasilinear degenerate diffusion equation with a source term depending solely on the gradient is investigated. After a suitable rescaling of time, convergence to a unique profile is shown for global solutions. The proof relies on the half-relaxed limits technique within the theory of viscosity solutions and on the construction of suitable supersolutions and barrier functions to obtain optimal temporal decay rates and boundary estimates. Blowup of weak solutions is also studied.
We consider equivariant wave maps from the $(d+1)$--dimensional Minkowski spacetime into the $d$-sphere for $d\geq 4$. We find a new explicit stable self-similar solution and give numerical evidence that it plays the role of a universal attractor for generic blowup. An analogous result is obtained for the $SO(d)$ symmetric Yang-Mills field for $d\geq 6$.
We prove quantitative regularity and blowup theorems for the incompressible Navier-Stokes equations in $\mathbb R^d$, $d\geq4$ when the solution lies in the critical space $L_t^\infty L_x^d$. Explicit subcritical bounds on the solution are obtained in terms of the critical norm. A consequence is that $\|u(t)\|_{L_x^d(\mathbb R^d)}$ grows at a minimum rate of $(\log\log\log\log(T_*-t)^{-1})^c$ along a sequence of times approaching a hypothetical blowup at $T_*$. These results quantify a theorem of Dong and Du and extend the three-dimensional work of Tao.
In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schrödinger equations $i\partial_tu+Δu+|u|^\frac4du=0$ in $\mathbb{R}^d$ with $d\geq3$. We aim to extend the result in [Colliander and Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Anna., 345(2009), 307-366.] for dimension two to the higher dimensions cases $d\geq3$, where we use the bootstrap argument in the above paper and the commutator estimates in [M. Visan and X. Zhang, O…
We study Smoluchowski-Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation several profiles of blowup solution have been noticed. Here we show the residual vanishing.
We establish the asymptotics of blowup for nonlinear heat equations with superlinear power nonlinearities in arbitrary dimensions and we estimate the remainders.
This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $ε$ regularity criterion, we show that if the mild solution $u$ with initial data in $\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d) $, $d<p,\,q<\infty$ becomes singular at a finite time $T_*$, then $$ \limsup_{t\to T_*} \|u(t)\|_{\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d)} = \infty. $$ The corresponding result in 3D case has been obtained by I.Gallagher, G.S.KochandF.Planchon. As a by-product, we also prove a regularity criterion for the Le…
It is shown that the one-dimensional or two-dimensional radially symmetric isothermal compressible Navier-Stokes system has no non-trivial global smooth solutions if the initial density is compactly supported. This result is a generalization of Xin's work \cite{Xin98} to the isothermal case.
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…
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.
Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic-parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting.
We study long time existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. In particular, we obtain exterior domain analogs of the four dimensional results of Hörmander where the nonlinearity is permitted to depend on the solution not just its first and second derivatives. Previous proofs in exterior domains omitted this dependence as it did not mesh well with the energy methods in use.
A simple proof of concentration of mass equal to $8π$ for blowing up $N$-symmetric solutions of the Keller--Segel model of chemotaxis in two dimensions with large $N$ is given. Moreover, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is derived for radial solutions of chemotaxis in several dimensions. This condition is, in a sense, complementary to the one guaranteeing the global-in-time existence of solutions.
We establish a blowup criterion for the two-dimensional (2D) full compressible Navier-Stokes system. The criterion is given in terms of the divergence of the velocity field only, and is independent of the temperature. It is almost the same as that of barotropic case.
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.
We consider the focusing inhomogeneous mass critical half-wave equation in one dimension. Under the mild conditions of the inhomogeneous factor, we show that the existence of the radial blowup solutions with ground state mass $\|u_0\|_2=\|Q\|_2$, where $Q$ is the unique positive ground state solution of equation $DQ+Q=Q^3$ and obtain the blowup rate $\|D^{\frac{1}{2}}u(t)\|_{L^2}\sim\frac{1}{|t|}$ as $t\nearrow 0^-$.
In this paper we use formal asymptotic arguments to understand the stability proper- ties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. We also analyze both the harmonic map heatflow and Schrodinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic. Solutions permitted to deviate from radial symmetry remain global for all time but may, for suitab…
We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a ring that shrinks to a point. In particular, the singularity is of type II, non self-similar. We show the stability of this dynamics among spherically symmetric solutions. In renormalised variables, the solution ressembles a traveling wave imploding at the origin, and this, to our knowledge, is the first stability result for such phenomenon for an evolution PD…
A key feature of $(1+1)$-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for $(1+1)$-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems.
We study the initial-boundary value problem of quasilinear wave equations outside of a star-shaped obstacle in four space dimensions, in which the nonlinear term under consideration may explicitly depend on the unknown function itself. By some new $L^{\infty}_{t}L^{2}_{x}$ and weighted $L^{2}_{t,x}$ estimates for the unknown function itself, together with energy estimates and KSS estimates, for the quasilinear obstacle problem we obtain a lower bound of the lifespan $T_{\varepsilon}\geq \exp{(\frac{c}{\varepsilon^2})}$, which coincides with the sharp lower bound of lifespan estimate for the co…
