Paschalis Karageorgis
We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential induces growth in the linearized problem, however, solutions that start out small may blow-up in finite time.
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Two families of sign-changing solutions of higher-order semilinear parabolic equations are studied.
We present a blow-up rate estimate for a solution to the parabolic Gross-Pitaevskii and related systems on entire space with Sobolev subcritical nonlinearity. We extend the results of [Y. Giga, S. Matsui and S. Sasayama, Indiana Univ. Math. J. {53} (2004), 483--514] to the parabolic systems.
We prove sharp pointwise blow-up estimates for finite-energy sign-changing solutions of critical equations of Schrödinger-Yamabe type on a closed Riemannian manifold $(M,g)$ of dimension $n \ge 3$. This is a generalisation of the so-called $C^0$-theory for positive solutions of Schrödinger-Yamabe type equations. To deal with the sign-changing case we develop a method of proof that combines an \emph{a priori} bubble-tree analysis with a finite-dimensional reduction, and reduces the proof to obtaining sharp \emph{a priori} blow-up estimates for a linear problem.
Let $Ω$ be a bounded domain in $\R^n$, $n\ge 3$ with smooth boundary $\partialΩ$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -Δu=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } Ω, \quad \quad u=0 \mbox{ on } \partial Ω, $$ where $\ve$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$ -Δw = |w|^{\frac{4}{n-2}} w, \ \ w \in {\mathcal D}^{1,2} (\R^n) $$ which has large number ($3n$) of kernels.
We consider the Brezis-Nirenberg problem: $$-Δu =λu + |u|^{p-1}u\qquad \mbox{in}\,\, Ω,\quad u=0\,\, \mbox{on}\,\,\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$, $p=\frac{N+2}{N-2}$ and $λ>0$. In this paper we prove that, if $Ω$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $λ\rightarrow λ_1$, where $λ_1=λ_1(Ω)$ denotes the first eigenvalue of $-Δ$ on $Ω$, with zero Dirichlet boundary condition.
Given a sufficiently symmetric domain $Ω\Subset\mathbb{R}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $β>4πk$ we construct blowing-up solutions $(u_\varepsilon)\subset H^1_0(Ω)$ to the Moser-Trudinger equation such that as $\varepsilon\downarrow 0$, we have $\|\nabla u_\varepsilon\|_{L^2}^2\to β$, $u_\varepsilon \rightharpoonup u_0$ in $H^1_0$ where $u_0$ is a sign-changing solution of the Moser-Trudinger equation and $u_\varepsilon$ develops $k$ positive spherical bubbles, all concentrating at $0\in Ω$. These $3$ features (lack of quantization, non-zero weak limit and bubble clustering) …
Given $(M,g)$ a compact Riemannian manifold of dimension $n\geq 3$, we are interested in the existence of blowing-up sign-changing families $(\ue)_{\eps>0}\in C^{2,θ}(M)$, $θ\in (0,1)$, of solutions to $$Δ_g \ue+h\ue=|\ue|^{\frac{4}{n-2}-\eps}\ue\hbox{ in }M\,,$$ where $Δ_g:=-\hbox{div}_g(\nabla)$ and $h\in C^{0,θ}(M)$ is a potential. We prove that such families exist in two main cases: in small dimension $n\in \{3,4,5,6\}$ for any potential $h$ or in dimension $3\leq n\leq 9$ when $h\equiv\frac{n-2}{4(n-1)}\Scal_g$. These examples yield a complete panorama of the compactness/noncompactness of…
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…
We investigate the blow-up behavior of sequences of sign-changing solutions for the Yamabe equation on a Riemannian manifold $(M,g)$ of positive Yamabe type. For each dimension $n\ge11$, we describe the value of the minimal energy threshold at which blow-up occurs. In dimensions $11 \le n \le 24$, where the set of positive solutions is known to be compact, we show that the set of sign-changing solutions is not compact and that blow-up already occurs at the lowest possible energy level. We prove this result by constructing a smooth, non-locally conformally flat metric on space forms $\mathbb{S}…
We consider the critical heat equation \begin{equation} \label{CH}\tag{CH} \begin{array}{lr} v_t-Δv =|v|^{\frac{4}{n-2}}v & Ω_ε\times (0, +\infty) \\ v=0 & \partialΩ_ε\times (0, +\infty) \\ v=v_0 & \mbox{ in } Ω_ε\times \{t=0\} \end{array} \end{equation} in $Ω_ε:=Ω\setminus B_ε(x_0)$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$ and $B_ε(x_0)$ is a ball of $\mathbb R^N$ of center $x_0\inΩ$ and radius $ε>0$ small. \\ We show that if $ε>0$ is small enough, then there exists a sign-changing stationary solution $ϕ_ε$ of \eqref{CH} such that the solution of \eqref{CH} with initia…
One-bubble nodal blow-up for asymptotically critical stationary Schrödinger-type equations
针对闭流形上渐近临界稳态非线性 Schrödinger 方程的单变号泡爆破，本文导出约束爆破点位置的新必要条件，揭示势函数、流形几何与泡之间的强相互作用。
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^n$ and denote the regular part of the Green's function on $Ω$ with Dirichlet boundary condition as $H(x,y)$. Assume that $q \in Ω$ and $n\geq 5$. We prove that there exists an integer $k_0$ such that for any integer $k\geq k_0$ there exist initial data $u_0$ and smooth parameter functions $ξ(t)\to q$, $0<μ(t)\to 0$ as $t\to +\infty$ such that the solution $u_q$ of the critical nonlinear heat equation \begin{equation*} \begin{cases} u_t = Δu + |u|^{\frac{4}{n-2}}u\text{ in } Ω\times (0, \infty),\\ u = 0\text{ on } \partial Ω\times (0, \infty),\\…
We construct blowing-up sign-changing solutions to some nonlinear critical equations by glueing a standard bubble to a degenerate function. We develop a method based on analyticity to perform the glueing when the critical manifold of solutions is degenerate and no Bianchi--Egnell type condition holds.
Blow-up rate for the subcritical semilinear heat equation in non-convex domains
本文研究非凸无界区域中亚临界半线性热方程的爆破行为，证明变号解不存在 II 型爆破，并推出尺度临界范数爆破，解决了 1980 年代以来的开放问题。
Infinite time blow-up and slow decay for the six dimensional energy-critical heat equation with self-similarly decaying initial data
研究六维能量临界半线性热方程中自相似衰减初始数据的解，发现存在符号变化的解在无限时间内爆破，以及衰减速度严格慢于自相似速率的非负解，证明爆破和衰减速率不唯一由初始数据衰减率决定。
Consider the nonlinear heat equation $v_t -Δv= |v|^{p-1} v$ in a bounded smooth domain $Ω\subset \R^n$ with $n>2$ and Dirichlet boundary condition. Given $u_{p}$ a sign-changing stationary solution fulfilling suitable assumptions, we prove that the solution with initial value $θu_{p} $ blows up in finite time if $ |θ-1|>0$ is sufficiently small and if $p$ is sufficiently close to the critical exponent. Since for $θ=1$ the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped. This phenomenon had been previously observed…
We consider the semilinear heat equation \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll}v_t-Δv= |v|^{p-1}v & \mbox{in}Ω\times (0,T)\\ v=0 & \mbox{on}\partial Ω\times (0,T)\\ v(0)=v_0 & \mbox{in}Ω\end{array}\right.\tag{$\mathcal P_p$} \end{equation} where $p>1$, $Ω$ is a smooth bounded domain of $\mathbb R^2$, $T\in (0,+\infty]$ and $v_0$ belongs to a suitable space. We give general conditions for a family $u_p$ of sign-changing stationary solutions of \eqref{problemAbstract}, under which the solution of \eqref{problemAbstract} with initial value $v_0=λu_p$ blows up in finite ti…
In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane-Emden-Fowler equations with singular nonlinearities and subquadratic convection term. We study the combined effects of various terms involved in these problems: sublinear or superlinear nonlinearities, singular nonlinear terms, convecti…
We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the numbe…
Stable critical point of the Robin function and bubbling phenomenon for a slightly subcritical elliptic problem
本文研究略次临界椭圆问题，证明当参数足够小时存在集中于Robin函数稳定临界点的变号气泡解，肯定回答了Clapp等人关于同点爆破变号解的猜想。
We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-Δu=|x|^α|u|^{p}+ζ(t)\,{\mathbf w}(x)$ in $(0,\infty)\times\mathbb{R}^N$, where $N\geq 3$, $p>1$, $α>-2$, $\z, {\mathbf w}$ are continuous functions such that $ζ(t)=t^σ$ or $ζ(t)\sim t^σ$ as $t\to 0$, $ζ(t)\sim t^m$ as $t\to\infty$ . We obtain local existence for $σ>-1$. We also show the following: \begin{itemize} \item If $m\leq 0$, $p<\frac{N-2m+α}{N-2m-2}$ and $\int_{\mathbb{R}^N}{\mathbf w}(x)dx>0$, then all solutions blow up in finite time; \item If $m> 0$, $p>1$ and $\int_{\mathbb{R}^N}{\math…
We prove the existence of multiple positive radial solutions to the sign-indefinite elliptic boundary blow-up problem \[ \left\{\begin{array}{ll} Δu + \bigl(a^+(\vert x \vert) - μa^-(\vert x \vert)\bigr) g(u) = 0, & \; \vert x \vert < 1, \\ u(x) \to \infty, & \; \vert x \vert \to 1, \end{array} \right. \] where $g$ is a function superlinear at zero and at infinity, $a^+$ and $a^-$ are the positive/negative part, respectively, of a sign-changing function $a$ and $μ> 0$ is a large parameter. In particular, we show how the number of solutions is affected by the nodal behavior of the weight functi…
In a recent paper D. D. Hai showed that the equation $ -Δ_{p} u = λf(u) \mbox{in} Ω$, under Dirichlet boundary condition, where $Ω\subset {\bf R^N}$ is a bounded domain with smooth boundary $\partialΩ$, $Δ_{p}$ is the p-Laplacian, $f : (0,\infty) \rightarrow {\bf R} $ is a continuous function which may blow up to $\pm \infty$ at the origin, admits a solution if $λ> λ_0$ and has no solution if $0 < λ< λ_0$. In this paper we show that the solution set $\mathcal{S}$ of the equation above, which is not empty by Hai's results, actually admits a continuum of positive solutions.
We investigate the local existence, finite time blow-up and global existence of sign-changing solutions to the inhomogeneous parabolic system with space-time forcing terms $$ u_t-Δu =|v|^{p}+t^σw_1(x),\,\, v_t-Δv =|u|^{q}+t^γw_2(x),\,\, (u(0,x),v(0,x))=(u_0(x),v_0(x)), $$ where $t>0$, $x\in \mathbb{R}^N$, $N\geq 1$, $p,q>1$, $σ,γ>-1$, $σ,γ\neq0$, $w_1,w_2\not\equiv0$, and $u_0,v_0\in C_0(\mathbb{R}^N)$. For the finite time blow-up, two cases are discussed under the conditions $w_i\in L^1(\mathbb{R}^N)$ and $\int_{\mathbb{R}^N} w_i(x)\,dx>0$, $i=1,2$. Namely, if $σ>0$ or $γ>0$, we show that the…
We consider radial solutions of the slightly subcritical problem $-Δu_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. A…
Existence and compactness of conformal metrics on the plane with unbounded and sign-changing Gaussian curvature
该研究针对平面上的预定高斯曲率方程，给出了存在指定总曲率解的充要条件，并分析了解在总曲率趋近不同临界值时的紧性与爆破行为。
Fujita-type blow-up for inhomogeneous semilinear heat equations with regularly varying forcing
利用正则变化理论，通过空间质量函数的变差指数刻画临界指数，建立了非齐次半线性热方程Fujita型爆破的统一框架，并证明了次临界范围全局解不存在及临界情况爆破。
We consider the semilinear diffusion equation $\partial$ t u = Au + |u| $α$ u in the half-space R N + := R N --1 x (0, +$\infty$), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ' , 0) = 0 for x ' $\in$ R N --1. We prove that if the symbol of the operator A is of order a|$ξ$| $β$ near the origin $ξ$ = 0, for some …
Existence of Kazdan-Warner equation with sign-changing prescribed function
本文研究了闭Riemann曲面上带变号规定函数的Kazdan-Warner方程，在特定条件下证明了其对应泛函极小值的存在性，推广了已有结果，并分析了临界点序列的blow-up行为及渐近恒等式。
In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit sphere, and the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together to blow up arguments, a variant of Pohozaev's type identity, a refinement of regularity results for this type operators, an…
We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz \cite{rabi} related to entire solutions of the Schrödinger equation.
We study the asymptotic behavior, as $λ\rightarrow 0$, of least energy radial sign-changing solutions $u_λ$, of the Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ B_1\\ u=0 & \hbox{on}\ \partial B_1, \end{cases} \end{equation*} where $λ>0$, $2^*=\frac{2n}{n-2}$ and $B_1$ is the unit ball of $\R^n$, $n\geq 7$. We prove that both the positive and negative part $u_λ^+$ and $u_λ^-$ concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover we show that suitable rescalings of $u_λ^+$ and $u_λ^-$ conve…
Compactness results for Sign-Changing Solutions of critical nonlinear elliptic equations of low energy
本文研究临界非线性椭圆方程低能变分解的紧致性，证明在n≤5时解集在C²中无条件紧致，在n≥7时h不为零时紧致，补充了Devillanova-Solimini的结果。
In this work we study the existence of nodal solutions for the problem $$ -Δu = λu e^{u^2+|u|^p} \text{ in }Ω, \; u = 0 \text{ on }\partial Ω, $$ where $Ω\subseteq \mathbb R^2$ is a bounded smooth domain and $p\to 1^+$. If $Ω$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^+$, when $Ω$ is an arbitrary domain and $λ$ is small enough. As far as we know, this is the first construction of sign-c…
We study the initial value problem for the nonlinear Schrödinger equation. We will prove that the blow-up of the L^{2}-norm of solutions with suitable initial data. We impose a condition related to the sign of the data but put no restriction on their size.
In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-…
This paper is concerned with the parabolic-elliptic Keller-Segel system with nonlinear diffusion and signal-dependent sensitivity \begin{align}\tag{KS}\label{system} \begin{cases} u_t=Δ(u+1)^m-\nabla\cdot(uχ(v)\nabla v),\quad &x\inΩ, t>0,\\ 0=Δv-v+u, &x\inΩ, t>0 \end{cases} \end{align} under homogeneous Newmann boundary conditions and initial conditions, where $Ω=B_R(0)\subset\mathbb{R}^N$ ($N\geq3,\ R>0$) is a ball, $m\geq 1$, $χ$ is a function satisfying that $χ(s)\geqχ_0(a+s)^{-k}$ ($k>0$, $χ_0>0$, $a\geq 0$) for all $s>0$ and some conditions. If the case that $m=1$ and $χ(s)=χ_0s^{-k}$, Na…
Blow-Up Results of Time Fractional Heat Equation With Nonlinear Neumann Boundary Conditions
本文研究具有幂型非线性Neumann边界条件的时间分数阶热方程，证明正解在有限时间内爆破，并确定爆破集仅位于边界点。
A critical nonlinearity for blow-up in a higher-dimensional chemotaxis system with indirect signal production
该研究针对3、4维球域上带间接信号产生的趋化系统Neumann问题，推导得出当反应项非线性增长阶σ>4/n时存在径向对称有限时间爆破解，明确了高维趋化系统爆破的临界非线性条件。
We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite-time blow-up solution with an isolated characteristic blow-up point at the origin, and a blow-up surface which is centered at the origin and has the shape of a stylized pyramid, whose edges follow the bisectrices of the axes in $R^2$. The blow-up surface is differentiable outside the bisecrtices. As for the asymptotic behavior in similariy variables, the solution converges to the classical one-dimensional soliton outside the bisectrices. On the bisectrices outside the or…
Let $G=(V,E)$ be a locally finite, connected and weighted graph. We prove that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ and uniform polynomial volume growth of degree $m$, all non-negative solutions of the equation $\partial_tu=Δu+u^{1+α}$ blow up in a finite time provided that $α=\frac{2}{m}$. We also consider the blow-up problem under certain conditions for volume growth and initial value. The obtained results provide a significant complement to the work by Lin and Wu in earlier paper.
In the present paper, we study small data blow-up of the semi-linear wave equation with a scattering dissipation term and a time-dependent mass term from the aspect of wave-like behavior. The Strauss type critical exponent is determined and blow-up results are obtained to both sub-critical and critical cases with corresponding upper bound lifespan estimates. For the sub-critical case, our argument does not rely on the sign condition of dissipation and mass, which gives the extension of the result in \cite{Lai-Sch-Taka18}. Moreover, we show the blow-up result for the critical case which is a ne…
Sharp conditions for fully nonlinear equations with $(n-1)$-Hessian matrix: Part II. $k$-Hessian case
该研究针对$1\leq k<n$的$k$-Hessian型完全非线性方程，建立$q<2$时整体下解存在的充要条件，刻画径向解的有限半径爆破并推导相关估计，拓展了此前行列式情形的研究。
We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution $u(x,t)$, the graph $x\mapsto T(x)$ of its blow-up points and $\SS\subset $ the set of all characteristic points, and show that the $\SS$ has an empty interior. Finally, given $x_0\in \SS$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons with alternate signs and that $T(x)$ forms a corner of angle $\frac π2$ at $x_0$.
Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities
本文研究了带组合非线性项的热方程，发现梯度项会导致正解的临界Fujita指数发生不连续跳跃现象，并进一步分析了变号解及非齐次问题的有限时间爆破条件与临界指数。
We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blow-up solution with a characteristic point, we refine the blow-up behavior first derived by Merle and Zaag. We also refine the geometry of the blow-up set near a characteristic point, and show that except may be for one exceptional situation, it is never symmetric with the respect to the characteristic point. Then, we show that all blow-up modalities predicted by those authors do occur. More precisely, given any integer $k\ge 2$ and $ζ_0 \in \m R$, we construct a blow-up solution with a character…
A Viscoelastic Wave Equation with Nonlinear Damping and Supercritical Variable-Exponent Logarithmic Sources
该研究针对带超临界增长的粘弹性波动方程，在特定指数条件下证明解的局部存在性、有限时间爆破与全局存在性，还得到不同阻尼范围的能量衰减估计及寿命上界。
In this paper, we are concerned with stable solutions , possibly unbounded and sign-changing, of some semi-linear elliptic problem with mixed nonlinear boundary conditions. We establish the nonexistence of stable solutions, the main methods used are the Pohozaev identity, monotonicity formula of solutions together with a blowing down sequence.
In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity $|u|^p$ for any $p>1$, under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.
On solutions to $-Δu = V u$ near infinity
研究势函数爆破速率条件下，方程 $-Δu =Vu$ 弱解在无穷远处的唯一延拓性质与变号行为。
We study the self-similar blow-up profiles associated to the following second order reaction-diffusion equation with strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $0<p<1$ and $σ>2(1-p)/(m-1)$. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for $m+p>2$ and $σ$ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when $p<1$. We moreover prove that, if the con…
Scalar behavior for a complex multi-soliton arising in blow-up for a semilinear wave equation
本文研究复值半线性波动方程在破裂时的标量行为，通过解决复值一阶Toda系统，揭示了解可分解为具有交替符号的有限多个孤子。
Concentration on the Boundary and Sign-Changing Solutions for a Slightly Subcritical Biharmonic Problem
本文研究轻微次临界四阶双调和方程，利用Lyapunov-Schmidt有限维约化方法证明在适当条件下，问题存在在边界点集中爆破的正解与变号解。
In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with mixed nonlinearities $a |u_t|^p+b |u|^q$, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and Glassey conjecture. In some cases, we obtain existence results, where the lower bound of the lifespan agrees with the upper bound in order. In addition, our results apply for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign co…
Fully sign-changing Nehari constraint vs sign-changing solutions of a competitive Schrödinger system
本文通过完全变号Nehari约束和Krasnosel'skii泛数，构造了竞争Schrödinger系统中能量递增的变号解，并证明其在极限系统中收敛。
Critical norm blow-up rates for the energy supercritical nonlinear heat equation
本研究无需径向对称等假设，首次分类能量超临界非线性热方程解的临界范数最优爆破速率，通过基于定量ε-正则性准则的新策略结合定量Carleman不等式得证，表明能量结构非必要，为其他抛物方程相关研究铺路。
In this article we will study the existence, multiplicity and Morse index of sign changing solutions for the Hardy-Sobolev-Maz'ya(HSM) equation in bounded domain and involving critical growth. We obtain infinitely many sign changing solutions for HSM equation. We also establish an estimate on the Morse index for the sign changing solutions.
In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equation {equation*} Δ^2 u=|x|^a |u|^{p-1}u\;\ {in}\;\ Ω, {equation*}where $a \ge 0$, $p>1$ and $Ω\subset \mathbb{R}^n$ is an unbounded domain of $\mathbb{R}^n$, $n \ge 5$. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the {\it Pohozaev-type identity}, {\it monotonicity formula} of solutions and a {\it blowing down} sequence, which is used …
We consider $u(x,t)$, a solution of $\partial_tu = Δu + |u|^{p-1}u$ which blows up at some time $T > 0$, where $u:\mathbb{R}^N \times[0,T) \to \mathbb{R}$, $p > 1$ and $(N-2)p < N+2$. Define $S \subset \mathbb{R}^N$ to be the blow-up set of $u$, that is the set of all blow-up points. Under suitable nondegeneracy conditions, we show that if $S$ contains a $(N-\ell)$-dimensional continuum for some $\ell \in \{1,\dots, N-1\}$, then $S$ is in fact a $\mathcal{C}^2$ manifold. The crucial step is to derive a refined asymptotic behavior of $u$ near blow-up. In order to obtain such a refined behavior,…
In this paper we study a zero-flux attraction-repulsion che\-mo\-taxis-system. We show that despite any mutual interplay between the repulsive and attractive coefficients from the corresponding chemo-sensitivities, even less any restriction on their own sizes, if the production rate of that chemical signal responsible of the cellular coalescence is sublinear, then any initial data emanate a unique global classical solution, which is as well bounded. Further, in a remark of the manuscript, we also address an open question given in \cite{Viglialoro2019RepulsionAttraction}.
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