We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space:
\begin{align}\label{abs:eqn} \left\{\begin{array}{ll} \partial_{t}u=Δ_{\mathbb{H}^{n}} u+ f(u, t) &\hbox{ in }~ \mathbb{H}^{n}\times (0, T),\\ \\ \quad u =u_{0} &\hbox{ in }~ \mathbb{H}^{n}\times \{0\}. \end{array}\right. \end{align}
We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, it exhibits Fujita phenomena for the exponential weight $h(t) = e^{μt},$ i.e. there exists a critical exponent $μ^*$ such that if $μ> μ^*$ then all non-negative solutions blow-up in finite time and if $μ\leq μ^*$ there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that the above mentioned Cauchy problem with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.
This paper studies of a variation of the hyperbolic blow up scenario suggested by Hou and Luo's recent numerical simulation [12]. In particular, we propose a "hyperbolic" surface quasi-geostrophic equation characterized by a incompressible velocity field with a modified Biot-Savart law. For this model, we will show Finite time blow up for a wide class of initial data.
In recent work of Luo and Hou, a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call "hyperbolic Boussinesq system". This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot-Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of…
We consider an initial boundary value problem for a 2x2 system of conservation laws modeling heatless adsorption of a gaseous mixture with two species and instantaneous exchange kinetics, close to the system of Chromatography. In this model the velocity is not constant because the sorption effect is taken into account. Exchanging the roles of the x, t variables we obtain a strictly hyperbolic system with a zero eigenvalue. Our aim is to construct a solution with a velocity which blows up at the corresponding characteristic "hyperbolic boundary" {t = 0}.
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…
We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blow-up dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones …
We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{μt}u^p$ ($μ\in\mathbb{R}, p>1$), posed on cones of the hyperbolic space. Under a certain assumption on $μ$ and $p$, related to the bottom of the spectrum of $-Δ$ in $\mathbb H^n$, we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters $μ$ and $p$ satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when …
We study the asymptotic behavior of blow-up solutions of the heat equation with nonlinear boundary conditions. In particular, we classify the asymptotic behavior of blow-up solutions and investigate the spacial singularity of their blow-up profiles.
In this article, we establish the existence of a family of hypersurfaces $(Γ(t))_{0< t \leq T}$ which evolve by the vanishing mean curvature flow in Minkowski space and which as $t$ tends to~$0$ blow up towards a hypersurface which behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second order quasilinear wave equation. Our constructive approach consists in proving the existence of finite time blow up solutions of this hyperbolic equation under the form $u(t,x) \sim t^ {ν+1} Q\Big(\frac {x} {t^ {ν+1}} \Big) $, where~$Q$ is a stationary …
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.
We consider the the pointwise estimates and the blow-up rate estimates for the zero Dirchilet problem of the semilinear heat equation with a gradient term.
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 finite blow-up solutions of the heat equation with nonlinear boundary conditions. It is known that a rate of blow-up solutions is the same as the self-similar rate for a Sobolev subcritical case. A goal of this paper is to construct a blow-up solution whose blow-up rate is different from the self-similar rate for a Sobolev supercritical case.
We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.
We establish a blow-up criterion in terms of the upper bound of the density and temperature for the strong solution to 2D compressible viscous heat-conductive flows. The initial vacuum is allowed.
Blowup analysis for solutions of a general evolution equation with nonlocal diffusion and localized source is performed. By comparison with recent results on global-in-time solutions, a dichotomy result is obtained.
We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work (Abdelhedi-Zaag JDE 2021), we constructed a blow-up solution for that equation, and showed that it blows up (at least) at the origin. We also derived the so called "intermediate blow-up profile". In this paper, we prove the single point blow-up property and determine the final blow-up profile.
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.
This note is devoted to a simple proof of blowup of solutions for a nonlinear heat equation. The criterion for a blowup is expressed in terms of a Morrey space norm and is in a sense complementary to conditions guaranteeing the global in time existence of solutions. The method goes back to H. Fujita and extends to other nonlinear parabolic equations.
We construct a solution for a class of strongly perturbed semilinear heat equations which blows up in finite time with a prescribed blow-up profile. The construction relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude.
We investigate the $p-$Laplace heat equation $u_t-Δ_p u=ζ(t)f(u)$ on a bounded smooth domain $Ω\subset\mathbb{R}^N$. Using differential inequalities arguments, we prove blow-up results under suitable conditions on $ζ, f$, and the initial data $u_0$. We also give an upper bound for the blow-up time in each case.
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.
The harmonic map heat flow is a geometric flow well known to produce solutions whose gradient blows up in finite time. A popular model for investigating the blow-up is the heat flow for maps $\mathbb R^{d}\to S^{d}$, restricted to equivariant maps. This model displays a variety of possible blow-up mechanisms, examples include self-similar solutions for $3\le d\le 6$ and a so-called Type II blow-up in the critical dimension $d=2$. Here we present the first constructive example of Type II blow-up in higher dimensions: for each $d\ge7$ we construct a countable family of Type II solutions, each ch…
Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of $\R^d$ are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.
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}.
In this paper, the initial and boundary problem of the difference equation which is a discretization of the semi-linear heat equation. The difference equation derived by discretizing the semi-linear heat equation has solutions which show characteristics corresponding to the characteristics of the blow-up solutions for the original equation. The initial and boundary problem for the original equation has blow-up solutions when a certain condition is met. We prove that when a similar condition as that of the original solution is met in the initial and boundary problem for the difference equation,…
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.
In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.
We analyze the finite-time blow-up of solutions of the heat flow for $k$-corotational maps $\mathbb R^d\to S^d$. For each dimension $d>2+k(2+2\sqrt{2})$ we construct a countable family of blow-up solutions via a method of matched asymptotics by glueing a re-scaled harmonic map to the singular self-similar solution: the equatorial map. We find that the blow-up rates of the constructed solutions are closely related to the eigenvalues of the self-similar solution. In the case of $1$-corotational maps our solutions are stable and represent the generic blow-up.
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.
We consider a nonlinear heat equation with a gradient term. We construct a blow-up solution for this equation with a prescribed blow-up profile. For that, we translate the question in selfsimilar variables and reduce the problem to a finite dimensional one. We then solve the finite dimensional problem using index theory. The interpretation of the finite dimensional parameters allows us to derive the stability of the constructed solution with respect to initial data.
We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier works [1, 2], we constructed a solution $u$ for that equation such that $u$ and $\nabla u$ both blow up at the origin and only there. We also gave the final blow-up profile. In this paper, we refine our construction method in order to get a sharper estimate on the gradient at blow-up.
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.
We consider the following Cauchy problem for three dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^{5},~&\mbox{ in } \ {\mathbb R}^3 \times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } \ {\mathbb R}^3. \end{cases} \end{equation*} We construct type II finite time blow-up solution $u(x,t)$ with the blow-up rates $ \| u\|_{L^\infty} \sim (T-t)^{-k}$, where $ k=1,2,... $. This gives a rigorous proof of the formal computations by Filippas, Herrero and Velazquez \cite{fhv}. This is the first instance of type II finite time blow-up for three dimensional energy critical heat…
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.
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 the present paper, we investigate blow-up and lifespan estimates for a class of semilinear hyperbolic coupled system in $\mathbb{R}^n$ with $n\geqslant 1$, which is part of the so-called Nakao's type problem weakly coupled a semilinear damped wave equation with a semilinear wave equation with nonlinearities of derivative type. By constructing two time-dependent functionals and employing an iteration method for unbounded multiplier with slicing procedure, the results of blow-up and upper bound estimates for the lifespan of energy solutions are derived. The model seems to be hyperbolic-like i…
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…
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 consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. We prove the existence of a blow-up solution, and give its blow-up profile. Our proof relies on the following method: we linearize the equation (in similarity variables) around the expected profile, then, we control the nonpositive directions of the spectrum thanks to the decreasing properties of the kernel. Finally, we use a topological argument to control the positive directions of the spectrum.
In this paper, the authors consider an initial boundary value problem for the heat flow of equation of surfaces with constant mean curvatures, which was investigated in [On the heat flow of equation of surfaces of constant mean curvatures, Manuscripta Mathematica, 2011, 134: 259-271] by Huang et al., where global well-posedness and finite time blow-up of regular solutions were obtained. Their results are complemented in this paper in the sense that some new conditions on the initial data are provided for the solutions to develop finite time singularity.
In this article, we study a semi-linear heat equation with the nonlinearity which is the product of polynomial and logarithmic functions. Using the invariance of the potential well(s), we have established the global existence and exponential decay estimates of solutions in $L^2$ - norm without having any restriction on the exponent in the source term under suitable conditions on the initial data. Moreover, finite time blow up of solutions at subcritical, critical and supercritical initial energy levels is also discussed.
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…
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. …
We consider the energy critical four dimensional semi-linear heat equation \[ \partial_{t}v-Δv-v^{3}=0, \quad(t,x)\in \mathbb{R}\times \mathbb{R}^4. \] Formal computation of Filippas et al. (R. Soc. Lond. Proc. 2000) conjectures the existence of a sequence of type II blow-up solutions with various blow-up rates \[ \|v(t)\|_{L^\infty(\mathbb{R}^4)}\approx \frac{|\log(T-t)|^{\frac{2L}{2L-1}}}{(T-t)^L} ,\quad L=1,2,\cdots.\] Schweyer (J. Funct. Anal. 2012) rigorously constructs a type II blow-up solution for the case $L=1$. In this paper, we show the existence of type II blow-up solution for $L=2…
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 this paper, we study the formation of finite time singularities in the form of super norm blowup for a spatially inhomogeneous hyperbolic system. The system is related to the variational wave equations as those in [18]. The system posses a unique $C^1$ solution before the emergence of vacuum in finite time, for given initial data that are smooth enough, bounded and uniformly away from vacuum. At the occurrence of blowup, the density becomes zero, while the momentum stays finite, however the velocity and the energy are both infinity.
