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 $d\ge 2$ . 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.     

On Stability of Pseudo-Conformal Blowup for L 2-critical Hartree NLS

We consider L ²-critical focusing nonlinear Schrödinger equations with Hartree type nonlinearity $i \partial_{t} u = - \Delta u - \left(\Phi \ast |u|^2 \right) u \quad {\rm in}\, \mathbb {R}^4,$ where Φ(x) is a perturbation of the convolution kernel |x|^?2. Despite the lack of pseudo-conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions u(t, x) that exhibit the pseudo-conformal blowup rate $\| \nabla u(t) \|_{L^2_x}\sim \frac{1}{|t|} \quad {\rm as}\, t \nearrow 0.$ Furthermore, we prove the finite-codimensional stability of this conformal blow up, by extending the nonlinear wave operator construction by Bourgain and Wang (see Bourgain and Wang in Ann. Scuola Norm Sup Pisa Cl Sci (4) 25(1–2), 197–215, 1997/1998) to L ²-critical Hartree NLS.     

Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces

We prove a weak comparison principle in narrow unbounded domains for solutions to $-\Delta _p u=f(u)$ in the case $2<p< 3$ and $f(\cdot )$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot )$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-\Delta _p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.     

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 nonlinear cross-diffusion terms. To this end we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller–Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.     

Borderline gradient continuity for nonlinear parabolic systems

We consider the evolutionary \(p\) -Laplacean system $$\begin{aligned} \partial _t u-\triangle _p u=F,\qquad p > \frac{2n}{n+2} \end{aligned}$$ in cylindrical domains of \( \mathbb R^{n}\times \mathbb R\) , and prove the continuity of the spatial gradient \(Du\) under the Lorentz space assumption \(F\in L(n+2,1)\) . When \(F\) is time independent the condition improves in \(F \in L(n,1)\) . This is the limiting case of a result of DiBenedetto claiming that \(Du\) is Hölder continuous when \(F \in L^{q}\) for \(q>n+2\) . At the same time, this is the natural nonlinear parabolic analog of a linear result of Stein, claiming the gradient continuity of solutions to the linear elliptic system \(\triangle u \in L(n,1)\) is continuous. New potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions.     

Stability of blowup for a parabolic p-Laplace equation with nonlinear source

In this paper, we mainly consider the stability of blowup of solutions for the p-Laplace equation with nonlinear source ${u_t = {div}(|\nabla u|^{p-2}\nabla u) + u^q,\;\;(x,t)\in\mathbb{R}^N \times (0,T)}$ , with the initial value ${u(x,0) = u_0(x) \geq 0}$ , where ${\|u_0 (x)\|_{L^\infty} \leq M}$ and T < ∞ is the blowup time. Under a small oscillation around the radial initial value, we can prove the solution blows up in finite time and obtain the blowup rate estimate of the form ${\|u(\cdot,t)\|_{L^\infty}\leq C(T-t)^{-\frac{1}{q-1}}}$ , where the constant C > 0 is dependent only on N, p, q, and the parameters q and p are expected to be ${p > 2, p-1 < q < \frac{Np}{(N-p)}_+ -1}$ .     

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow-up scenario for the 3D Euler equation proposed by Hou and Luo (Multiscale Model Simul 12:1722–1776, 2014) based on extensive numerical simulations. These models generalize the 1D Hou–Luo model suggested in Hou and Luo Luo and Hou (2014), for which finite-time blowup has been established in Choi et al. (arXiv preprint. arXiv:1407.4776, 2014). The main new aspects of this work are twofold. First, we establish finite-time blowup for a model that is a closer approximation of the three-dimensional case than the original Hou–Luo model, in the sense that it contains relevant lower-order terms in the Biot–Savart law that have been discarded in Hou and Luo Choi et al. (2014). Secondly, we show that the blow-up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou–Luo model. Such blow-up stability result may be useful in further work on understanding the 3D hyperbolic blow-up scenario.     

Bounds for the blowup time of the solutions to quasi-linear parabolic problems

In this paper, we obtain the lower and the upper bounds of the blowup time of the solutions to quasi-linear parabolic problems subject to Dirichlet(or Neumann) boundary condition. Our results are suitable for the problems with any smooth bounded domain ${\Omega \subset \mathbb{R}^n}$ and ${n \geq 3}$ . In some special cases, we can even get the exact values of blowup time.     

Positive solution to semilinear parabolic equation associated with critical Sobolev exponent

We study the semilinear parabolic equation ${u_{t}- \Delta u = u^{p}, u \geq 0}$ on the whole space R ^N , ${N \geq 3}$ associated with the critical Sobolev exponent p = (N + 2)/(N ? 2). Similarly to the bounded domain case, there is threshold blowup modulus concerning the blowup in finite time. Furthermore, global in time behavior of the threshold solution is prescribed in connection with the energy level, blowup rate, and symmetry.     

A perturbation method for spinorial Yamabe type equations on S^m and its application

For $m\ge 2$ , we prove the existence of non-trivial solutions for a certain kind of nonlinear Dirac equations with critical Sobolev nonlinearities on $S^m$ via a perturbative variational method. For the special case $m=2$ , this establishes the existence of a conformal immersion $S^2\rightarrow \mathbb R ^3$ with prescribed mean curvature $H$ which is close to a positive constant under an index counting condition on $H$ .     

Gradient blowup rate for a viscous Hamilton–Jacobi equation with degenerate diffusion

This paper is concerned with the gradient blowup rate for the one-dimensional p-Laplacian parabolic equation ${u_t=(|u_x|^{p-2} u_x)_x +|u_x|^q}$ with q > p > 2, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish the blowup rate estimates of lower and upper bounds and show that in this case the blowup rate does not match the self-similar one.     

Generalized nonlinear complementarity problems with order P_0 and R_0 properties

We generalize the \(P, P_0, R_0\) properties for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of generalized nonlinear complementarity problem (GNCP). We prove that if a continuous function has the order \(P_0\) and \(R_0\) properties then all the associated GNCPs have solutions.     

Existence and non-existence of global solutions for uniformly parabolic equations

In this paper, we prove the existence of Fujita-type critical exponents for x-dependent fully nonlinear uniformly parabolic equations of the type $$(*)\quad \partial_{t}u=F(D^{2}u,x)+u^{p}\quad{\rm in}\ \ \mathbb{R}^{N}\times\mathbb{R}^{+}.$$ These exponents, which we denote by p(F), determine two intervals for the p values: in ]1,p(F)[, the positive solutions have finite-time blow-up, and in ]p(F), +∞[, global solutions exist. The exponent p(F)?=?1?+?1/α(F) is characterized by the long-time behavior of the solutions of the equation without reaction terms $$\partial_{t}u=F(D^{2}u,x)\quad{\rm in}\ \ \mathbb{R}^{N}\times\mathbb{R}^{+}.$$ When F is a x-independent operator and p is the critical exponent, that is, p?=?p(F). We prove as main result of this paper that any non-negative solution to (*) has finite-time blow-up. With this more delicate critical situation together with the results of Meneses and Quaas (J Math Anal Appl 376:514–527, 2011), we completely extend the classical result for the semi-linear problem.     

The Oldroyd-\alpha model for the incompressible viscoelastic flow

In this paper, we introduce and study a new model, which is called the Lagrangian-averaged Oldroyd- $\alpha $ (LAO- $\alpha $ ) model in two space dimensions. Such a model is inspired by the Lagrangian-averaged Navier–Stokes- $\alpha $ model (also known as the viscous Camassa–Holm equations). We obtain global existence result for the Cauchy problem of the LAO- $\alpha $ model. And we prove that a subsequence of solutions of the LAO- $\alpha $ equations converges to certain solution of the two-dimensional Oldroyd model as $\alpha $ converges to zero.     

Scalarization of \epsilon -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces

In this paper, we investigate the scalarization of \(\epsilon \) -super efficient solutions of set-valued optimization problems in real ordered linear spaces. First, in real ordered linear spaces, under the assumption of generalized cone subconvexlikeness of set-valued maps, a dual decomposition theorem is established in the sense of \(\epsilon \) -super efficiency. Second, as an application of the dual decomposition theorem, a linear scalarization theorem is given. Finally, without any convexity assumption, a nonlinear scalarization theorem characterized by the seminorm is obtained.     

Existence of entire solutions for a class of quasilinear elliptic equations

The paper deals with the existence of entire solutions for a quasilinear equation ${(\mathcal E)_\lambda}$ in ${\mathbb{R}^N}$ , depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ* > 0 with the property that ${(\mathcal E)_\lambda}$ admits nontrivial non-negative entire solutions if and only if λ ≥ λ*. Furthermore, when ${\lambda > \overline{\lambda} \ge \lambda^*}$ , the existence of a second independent nontrivial non-negative entire solution of ${(\mathcal{E})_\lambda}$ is proved under a further natural assumption on A.     

Everywhere regularity of certain nonlinear diffusion systems

In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the form $$\mathbf{u}_t -\Delta\left[ \mathbf{\nabla}\Phi(\mathbf{u})\right] = 0,$$ where ${\Phi(z)}$ is a strictly convex function. I show that when ${\Phi}$ is a function only of the norm of u, then bounded weak solutions of these parabolic systems are everywhere Hölder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner (Math Ann 292(4):711–727, 1992) on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems.     

Well-posedness and blowup phenomena for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs

This paper deals with the Cauchy problem for a cross-coupled Camassa–Holm equation $$m_t=-(vm)_x-mv_x, n_t=-(un)_x-nu_x,$$ where \({n\doteq v-v_{xx}}\) , \({m\doteq u-u_{xx}+\omega}\) with a constant ω. The local well-posedness of solutions for the Cauchy problem of the cross-coupled Camassa–Holm equation in Sobolev space \({H^s(\mathbb{R})}\) with s > 5/2 is established. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and the blowup scenario of the solutions to the equation is also obtained.     

On the heat flow on metric measure spaces: existence, uniqueness and stability

We prove existence and uniqueness of the gradient flow of the Entropy functional under the only assumption that the functional is λ-geodesically convex for some ${\lambda\in\mathbb {R}}$ . Also, we prove a general stability result for gradient flows of geodesically convex functionals which Γ?converge to some limit functional. The stability result applies directly to the case of the Entropy functionals on compact spaces.     

The local circular law III: general case

In the first part (Bourgade et al., Local circular law for random matrices, preprint, arXiv:1206.1449, 2012) of this article series, Bourgade, Yau and the author of this paper proved a local version of the circular law up to the finest scale \(N^{-1/2+ {\varepsilon }}\) for non-Hermitian random matrices at any point \(z \in \mathbb {C}\) with \(||z| - 1| > c \) for any constant \(c>0\) independent of the size of the matrix. In the second part (Bourgade et al., The local circular law II: the edge case, preprint, arXiv:1206.3187, 2012), they extended this result to include the edge case \( |z|-1={{\mathrm{o}}}(1)\) , under the main assumption that the third moments of the matrix elements vanish. (Without the vanishing third moment assumption, they proved that the circular law is valid near the spectral edge \( |z|-1={{\mathrm{o}}}(1)\) up to scale \(N^{-1/4+ {\varepsilon }}\) .) In this paper, we will remove this assumption, i.e. we prove a local version of the circular law up to the finest scale \(N^{-1/2+ {\varepsilon }}\) for non-Hermitian random matrices at any point \(z \in \mathbb {C}\) .