设$\mathbb{H}^n$为n维实双曲空间，$Δ$为其非负的拉普拉斯-贝尔特里算子，其谱底记为$λ_{0}$，$σ∈(0,1)$。本文旨在双曲空间上分数热方程$\partial_{t} u + Δ^σu = e^{βt}|u|^{γ-1}u$的Fujita指数进行研究，证明非平凡正全局解存在的充要条件为$γ≥1 + β/ λ_{0}^σ$。同时，证明了对于$0≤λ≤λ_{0}$和$1<γ< \frac{n+2σ}{n-2σ}$，半线性分数椭圆方程$Δ^σ v - λ^σ v - v^γ=0$存在非负、有界且有限能量的解。两个问题相互关联，后者除独立价值外，对前者至关重要。本文的核心结果是一个新的分数Poincaré型不等式，以新的$L^{2}$分数Sobolev空间形式表达，比现有结果更精细，并且更广泛地适用于非紧致的黎曼对称空间。还建立了径向函数的相关Rellich-Kondrachov型紧嵌入定理及其他相关性质。
英文摘要：
Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $Δ$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $λ_{0}$, and $σ\in (0,1)$.
The aim of this paper is twofold. On the one hand, we determine the Fujita exponent for the fractional heat equation \[\partial_{t} u + Δ^σu = e^{βt}|u|^{γ-1}u,\] by proving that nontrivial positive global solutions exist if and only if $γ\geq 1 + β/ λ_{0}^σ$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation \[
Δ^σ v - λ^σ v - v^γ=0 \] for $0\leq λ\leq λ_{0}$ and $1<γ< \frac{n+2σ}{n-2σ}$. The two problems are known to be connected and the latter, aside from its independent interest, is actually instrumental to the former.
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At the core of our results stands a novel fractional Poincaré-type inequality expressed in terms of a new scale of $L^{2}$ fractional Sobolev spaces, which sharpens those known so far, and which holds more generally on Riemannian symmetric spaces of non-compact type. We also establish an associated Rellich--Kondrachov-like compact embedding theorem for radial functions, along with other related properties.
This paper is concerned with finite blow-up solutions of a one dimensional complex-valued semilinear heat equation. We provide locations and the number of blow-up points from the viewpoint of zeros of the solution.
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It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is global in time and we give an example when it never becomes a classical solution again.
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We give blow-up results for the Klein-Gordon equation and other perturbations of the semilinear wave equations with superlinear power nonlinearity, in one space dimension or in higher dimension under radial symmetry outside the origin.
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In this paper we consider a semilinear parabolic equation with nonlinear and nonlocal boundary condition and nonnegative initial datum. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.
In this paper we prove the blow-up theorem in the critical case for weakly coupled systems of semilinear wave equations in high dimensions. The upper bound of the lifespan of the solution is precisely clarified.
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We introduce a straightforward method to analyze the blow-up of systems of ordinary differential inequalities, and apply it to study the blow- up of solutions to a weakly coupled system of semilinear heat equations. We prove that the solution blows up in a finite time under the subcritical condition. Moreover, we give estimates of lifespan of the solution and lower estimates of the blow-up rate for a localized average of each component.
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 derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.
Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type $$ u_t= -Δ^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, $$ are discussed. For the semilinear heat equation $u_t= Δu+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.
We consider a blow-up solution for the semilinear wave equation in $N$ dimensions, with subconformal power nonlinearity. Introducing $\RR_0$ the set of non-characteristic points with the Lorentz transform of the space-independent solution as asymptotic profile, we show that $\RR_0$ is open and that the blow-up surface is of class $C^1$ on $\RR_0$. Then, we show the stability of $\RR_0$ with respect to initial data.
The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=Δu+V(x) |u|^{p-1}u$ in $Ω\times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data $u(x,0)=M\vf (x)$ as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.
In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type $|u_t|^p$ in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow-up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent $p$ for the nonlinear term satisfies $1<p\leqslant (n+1)/(n-1)$ for $n\geqslant2$ and $p>1$ for $n=1$. In particular, we find the same blow-up range for $p$ as in the corresponding semilinear wave …
We consider in this work some class of strongly perturbed for the semilinear heat equation with Sobolev sub-critical power nonlinearity. We first derive a Lyapunov functional in similarity variables and then use it to derive the blow-up rate. We also classify all possible asymptotic behaviors of the solution when it approaches to singularity. Finally, we describe precisely the blow-up profiles corresponding to these behaviors.
Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be type II in critical case. When p>p_S=(N+2)/(N-2) for N>=3, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than N-2-4/(p-1), which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collaps…
In this paper, we study weakly coupled systems for semilinear wave equations with distinct nonlinear memory terms in general forms, and the corresponding single semilinear equation with general nonlinear memory terms. Thanks to Banach's fixed point theorem, we prove local (in time) existence of solutions with the $L^1$ assumption on the memory kernels. Then, blow-up results for energy solutions are derived applying iteration methods associated with slicing procedure. We investigate interactions on the blow-up conditions under different decreasing assumptions on the memory term. Particularly, a…
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
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We consider a nonlinear wave equation with nonconstant coefficients. In particular, the coefficient in front of the second order space derivative is degenerate. We give the blow-up behavior and the regularity of the blow-up set. Partial results are given at the origin, where the degeneracy occurs. Some nontrivial obstacles, due to the nonconstant speed of propagation, have to be surmounted.
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We consider initial value problem for semilinear damped wave equations in three space dimensions. We show the small data global existence for the problem without the spherically symmetric assumption and obtain the sharp lifespan of the solutions. This paper is devoted to a proof of the Takamura's conjecture on the lifespan of solutions.
We construct a solution to a complex nonlinear heat equation which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. We note that the real and imaginary parts of the constructed solution blow up simultaneously.
The blow-up for semilinear wave equations with the scale invariant damping has been well-studied for sub-Fujita exponent. However, for super-Fujita exponent, there is only one blow-up result which is obtained in 2014 by Wakasugi in the case of non-effective damping. In this paper we extend his result in two aspects by showing that: (I) the blow-up will happen for bigger exponent, which is closely related to the Strauss exponent, the critical number for non-damped semilinear wave equations; (II) such a blow-up result is established for a wider range of the constant than the known non-effective …
We consider in this paper some class of perturbation for the semilinear wave equation with subcritical (in the conformal transform sense) power nonlinearity. We first derive a Lyapunov functional in similarity variables and then use it to derive the blow-up rate. Though the result is similar to the unperturbed case in its statements, this is not the case of our method, which is new up to our knowledge.
We consider the Cauchy problem for semilinear wave equations with variable coefficients and time-dependent scattering damping in $\mathbf{R}^n$, where $n\geq 2$. It is expected that the critical exponent will be Strauss' number $p_0(n)$, which is also the one for semilinear wave equations without damping terms. Lai and Takamura (2018) have obtained the blow-up part, together with the upper bound of lifespan, in the sub-critical case $p<p_0(n)$. In this paper, we extend their results to the critical case $p=p_0(n)$. The proof is based on Wakasa and Yordanov (2018), which concerns the blow-up an…
We consider in this work some class of strongly perturbed for the semilinear wave equation with conformal power nonlinearity. We obtain an optimal estimate for a radial blow-up solution and we have also obtained two less stronger estimates. These results are achieved in three-steps argument by the construction of a Lyapunov functional in similarity variables and the Pohozaev identity.
We consider the semilinear heat equation $u_t=Δu+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction property. We show the existence of a new type of blowup solutions by using this property. In fact, we obtain such blowup solutions by connecting a specific blowup solution of $u_t=Δu+|u|^{p-1}u$ and a specific solution of $u_t=Δu-|u|^{q-1}u$, and by adding correction terms.
In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.
In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in $\mathbb{R}^n$, which can be represented by the Riemann-Liouville fractional integral of order $1-γ$ with $γ\in(0,1)$. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of $γ$, on the blow-up range between the effective case and the non-effective ca…
The paper investigates a class of a semilinear wave equation with time-dependent damping term ($-\frac{1}{{(1+t)}^β}Δu_t$) and a nonlinearity $|u|^p$. We will show the influence of the the parameter $β$ in the blow-up results under some hypothesis on the initial data and the exponent $p$ by using the test function method. We also study the local existence in time of mild solution in the energy space $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$.
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…
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.
We consider in this paper a large class of perturbed semilinear wave equation with subconformal power nonlinearity. In particular, we allow log perturbations of the main source. We derive a Lyapunov functional in similarity variables and use it to derive the blow-up rate. Throughout this work, we use some techniques developed for the unperturbed case studied by Merle and Zaag [12] together with ideas introduced by Hamza and Zaag in [5] for a class of weather perturbations.
We consider in this paper some class of perturbation for the semilinear wave equation with critical (in the conformal transform sense) power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using a two-step argument based on interpolation and a critical Gagliardo-Nirenberg inequality, we show that the blow-up rate of any sigular solution is given by the solution of the non perturbed associated ODE, namely $u" = u^p$.
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 $x_0$.
We consider the semilinear wave equation in the radial case with conformal subcritical power nonlinearity. If we consider a blow-up point different from the origin, then we exhibit a new Lyapunov functional which is a perturbation of the one dimensional case and extend all our previous results known in the one-dimensional case. In particular, we show that the blow-up set near non-zero non-characteristic points is of class $C^1$, and that the set of characteristic points is made of concentric spheres in finite number in $\{\frac 1R \le |x|\le R\}$ for any $R>1$.
In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.
Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning the non-existence of positive solutions in the half space. The regularity of solutions are also investigated.
We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this Lyapunov functional, we derive the blow-up rate and the blow-up limit of the solution. We also classify all asymptotic behaviors of the solution at the singularity and give precisely blow-up profiles corresponding to these behaviors. Finally, we attain the blow-up profile numerically, thanks to a new mesh-refinement algorithm inspired by the rescaling method of Be…
We consider the semilinear heat equation $$\partial_t u -Δu =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u' =|u|^{p-1}u\log^a (2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowl…
The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $Δ^{\fracα{2}} u=ρ(x)\,φ(u)$ admits nonnegative nontrivial bounded solutions in the whole space $\mathbb{R}^N$.
We consider the initial value problem for the semilinear wave equation with time-dependent effective damping. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the damping as $t\to \infty$. The result of this paper is the sharp lifespan estimates of blowup solutions for general time-dependent damping including threshold cases between effective and overdamping.
We consider the semilinear wave equation in higher dimensions with power nonlinearity in the super-conformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up solutions previously obtained by Killip, Stovall and Vişan [6]. Our proof uses the similarity variables' setting. We consider the equation in that setting as a perturbation of the conformal case, and we handle the extra terms thanks to the ideas we already developed in [5] for perturbations of the pure power case with lower order terms.
