We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based on comparison with global solutions. For a critical power non-linearity we obtain a two-parameter family of radially symmetric stationary solutions.
By extending the method of moving planes to fractional powers of the Laplacian we prove that all positive steady states of the corresponding equation in a finite ball are radially symmetric.
We prove the existence of a critical Fujita exponent for a non-homogeneous semilinear heat equation which involves degenerate coefficients. More precisely, in order to give a rather complete theory, we focus on two types of weights $w(x)=|x_1|^a$ or $w(x)=|x|^b$ where $a, b>0$ in a suitable range. The coefficients under consideration admit either a singularity at the origin or a line of singularities. In the latter case, the problem is related to the fractional Laplacian.
In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the Fractional Cauchy problem with the Hardy potential, namely, \begin{equation*} u_t+(-Δ)^s u=λ\dfrac{u}{|x|^{2s}}+u^{p}\inn\ren,\\ u(x,0)=u_{0}(x)\inn\ren, \end{equation*} where $N> 2s$, $0<s<1$, $(-Δ)^s$ is the fractional laplacian of order $2s$, $ł>0$, $u_0\ge 0$, and $1<p<p_{+}(s,λ)$, where $p_{+}(λ, s)$ is the critical existence power found in \cite{BMP} and \cite{AMPP}.
We consider non-negative solutions to the semilinear space-fractional diffusion problem $(\partial_t+(-Δ)^{α/2})u=ρ(x)u^p$ on whole space $R^n$ with nonnegative initial data and with $(-Δ)^{α/2}$ being the $α$-Laplacian operator, $α\in (0,2)$. Here $p>0$ and $ρ(x)$ is a non-negative locally integrable function. For $ρ(x)=1$ we show that the fujita exponent is $p_F=1+\fracα{n}$ and the Liouville type result for the stationary equation is true for $0<p\leq 1+\fracα{n-α}$. When $p=1/2$ and $ρ(x)$ satisfies an integrable condition, there is at least one positive solution. This existence result is …
We first study the so-called Heat equation with two families of elliptic operators whichare fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equationwith operators including the "large" eigenvalues has strong similarities with a Heatequation in lower dimension whereas, surprisingly, for operators including "small" eigenvalues it shares some properties with some transport equations. In particular, forthese operators, the Heat equation (which is nonlinear) not only does not have theproperty that "disturbances propagate with infinite speed" but may lead to quenchingin fi…
We shall prove dispersive and smoothing estimates for Bochner type laplacians on some non-compact Riemannian manifolds with negative Ricci curvature, in particular on hyperbolic spaces. These estimates will be used to prove Fujita-Kato type theorems for the incompressible Navier-Stokes equations. We shall also discuss the uniqueness of Leray weak solutions in the two dimensional case.
We establish a priori Lipschitz estimates for unbounded solutions of second-order Hamilton-Jacobi equations in R^N in presence of an Ornstein-Uhlenbeck drift. We generalize the results obtained by Fujita, Ishii \& Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a nonlocal integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear. These results are obtained for both degenerate …
In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold $M$: \begin{equation*} \begin{cases} u_{t}-\mathfrak{L}_{M} u=f(u), \;x\in M, \;t>0, \\u(0,x)=u_{0}(x), \;x\in M, \end{cases} \end{equation*} for $u_{0}\geq 0$, where $\mathfrak{L}_{M}$ is a sub-Laplacian of $M$. In the case when $M$ is a connected unimodular Lie group $\mathbb G$, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with $u_{0}\not\equiv 0$, blow up in finite time if and only if $1<p\leq p_…
We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.
Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent
Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the $p$-Laplacian operator, $p\ge 2$, and the source term a power of the norm of the gradient of $u$. As a first step, the radially symmetric and non-increasing stationary solutions are characterized.
In this paper we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group $\mathbf{H}_n$. The heat operator is given in this case by $\partial_t-Δ_H$, where $Δ_H$ is the so-called sub-Laplacian on $\mathbf{H}_n$. We prove that the Fujita exponent $1 + 2/Q$ is critical, where $Q = 2n + 2$ is the homogeneous dimension of $\mathbf{H}_n$. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical…
We investigate the boundedness and large time behavior of solutions of the Cauchy-Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity: $$ u_t = (|u-x|^{p-2} u-x)_x+|u_x|^q \qquad \text{in}\quad (0, +\infty)\tiles(0, 1),\qquad q > p > 2.$$ We prove that: either $u_x$ blows up in finite time, or $u$ is global and converges in $W^{1, \infty}$norm to the unique steady state. This in particular eliminates the possibility of global solutions with unbounded gradient. For that purpose a Lyapunov functional is constructed by the approach of Zelenyak.
We study a class of nonlocal reaction-diffusion equations with a harvesting term where the nonlocal operator is given by a Bernstein function of the Laplacian. In particular, it includes the fractional Laplacian, fractional relativistic operators, sum of fractional Laplacians of different order etc. We study existence, uniqueness and multiplicity results of the solutions to the steady state equation. We also consider the parabolic counterpart and establish the long time asymptotic of the solutions. Our proof techniques rely on both analytic and probabilistic arguments.
The question studied here is the existence and uniqueness of a non-trivial bounded steady state of a Fisher-KPP equation involving a fractional Laplacian (--$Δ$)^$α$ in a domain with Dirichlet conditions outside of the domain. More specifically, we investigate such questions in the case of general fragmented unbounded domains. Indeed, we take advantage of the non-local dispersion in order to provide analytic bounds (which depend only on the domain) on the steady states. Such results are relevant in biology. For instance, our results provide criteria on the domain for the subsistence of a speci…
Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical and numerical approaches. The purpose of this work is to study the effects of the spectral fractional Laplacian on the bifurcation structure of reaction-diffusion systems on bounded domains. In order to do this we use advanced numerical continuation techniques to compute the solution branches. Since current available continuation packages only support systems involving the standard Laplacian, we first…
We prove existence and uniqueness of strong (pointwise) solutions to a linear nonlocal strongly coupled hyperbolic system of equations posed on all of Euclidean space. The system of equations comes from a linearization of a nonlocal model of elasticity in solid mechanics. It is a nonlocal analogue of the Navier-Lamé system of classical elasticity. We use a well-known semigroup technique that hinges on the strong solvability of the corresponding steady-state elliptic system. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to co…
We consider the steady fractional Schrödinger equation $L u + V u = f$ posed on a bounded domain $Ω$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-Δ)^s$; $V\ge 0$ is a potential with possible singularities, and the right-hand side are integrable functions or Radon measures. We reformulate the problem via the Green function of $(-Δ)^s$ and prove well-posedness for functions as data.If $V$ is bounded or mildly singular a unique solution of $(-Δ)^s u + V u = μ$ exists for every Borel measure $μ$. On the other hand, when $V$ is allowed to be more …
