We consider the Cauchy-problem for a parabolic equation of the following type:
\begin{equation*}
\frac{\partial u}{\partial t}= Δu+ f(u,|x|),
\end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial condition $u(x,0)=ϕ$, and we allow $ϕ$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t,x;ϕ)$ for a wide class of non-homogeneous non-linearities $f$. We show that in the supercritical case, Ground States with slow decay lie on the threshold between blowing up initial data and the basin of attraction of the null solution. Our results extend previous ones allowing Matukuma-type potential and more generic dependence on $u$.
Then, we further explore such a threshold in the subcritical case too. We find two families of initial data $ζ(x)$ and $ψ(x)$ which are respectively above and below the threshold, and have arbitrarily small distance in $L^{\infty}$ norm, whose existence is new even for $f(u,r)=u^{q-1}$. Quite surprisingly both $ζ(x)$ and $ψ(x)$ have fast decay (i.e. $\sim |x|^{2-n}$), while the expected critical asymptotic behavior is slow decay (i.e. $\sim |x|^{2/q-2}$).
The nonnegative viscosity solutions to the infinite heat equation with homogeneous Dirichlet boundary conditions are shown to converge as time increases to infinity to a uniquely determined limit after a suitable time rescaling. The proof relies on the half-relaxed limits technique as well as interior positivity estimates and boundary estimates. The expansion of the support is also studied.
We investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions. It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infinity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.
We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex-valued functions on the square lattice in the way that the classical heat kernel arises in the (local) central limit theorem. These so-called positive-homogeneous operators generalize the class of semi-elliptic operators in the sense that the definition is coordinate-free. More generally, we introduce a class of …
The aim of this paper is to prove the existence and several selected properties of a global fundamental Heat kernel $Γ$ for the parabolic operators $\mathcal{H}=\sum_{j=1}^m X_j^2-\partial_t$, where $X_1,\ldots,X_m$ are smooth vector fields on $\mathbb{R}^n$ satisfying Hörmander'snrank condition, and enjoying a suitable homogeneity assumption with respect to a family of non-isotropic dilations. The proof of the existence of $Γ$ is based on a (algebraic) global lifting technique, together with a representation of $Γ$ in terms of the integral (performed over the lifting variables) of the Heat ke…
In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.
A system of segregated boundary-domain integral equations (BDIEs) is obtained from the Dirichlet problem for the diffusion equation in non-homogeneous media defined on an exterior two-dimensional domain. We use a parametrix different from the one employed by in (Dufera, 2019). The system of BDIEs is formulated in terms of parametrix-based surface and volume potentials whose mapping properties are analysed in weighted Sobolev spaces. The system of BDIEs is shown to be equivalent to the original boundary value problem and uniquely solvable in appropriate weighted Sobolev spaces suitable for infi…
Let $\mathcal{H}=\sum_{j=1}^{m}X_{j}^{2}-\partial_{t}$ be a heat-type operator in $\mathbb{R}^{n+1}$, where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth Hörmander's vector fields in $\mathbb{R}^{n}$, and every $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$, while no underlying group structure is assumed. In this paper we prove global (in space and time) upper and lower Gaussian estimates for the heat kernel $Γ(t,x;s,y)$ of $\mathcal{H}$, in terms of the Carnot-Carathéodory distance induced by $X$ on $\mathbb{R}^{n}$, as well as …
We consider a non--homogeneous incompressible and heat conducting fluid confined to a 3D domain perforated by tiny holes. The ratio of the diameter of the holes and their mutual distance is critical, the former being equal to $ε^3$, the latter proportional to $ε$, where $ε$ is a small parameter. We identify the asymptotic limit for $ε\to 0$, in which the momentum equation contains a friction term of Brinkman type determined uniquely by the viscosity and geometric properties of the perforation. Besides the inhomogeneity of the fluid, we allow the viscosity and the heat conductivity coefficient …
We consider the inhomogeneous heat equation on the half-space $\mathbb R_{+}^{d}$ with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol in [10].
We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum, and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain.
This paper concerns the final value problem for the heat equation under the homogeneous Neumann condition on the boundary of a smooth open set in Euclidean space. The problem is here shown to be isomorphically well posed in the sense that there exists a linear homeomorphism between suitably chosen Hilbert spaces containing the solutions and the data, respectively. This improves a recent work of the author, in which the problem was proven well-posed in the original sense of Hadamard under an additional assumption of Hölder continuity of the source term. The point of departure is an abstract ana…
We study homogeneous Besov and Triebel--Lizorkin spaces defined on doubling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. When the measure space is a smooth manifold and such operator is a sum of squares of smooth vector fields, we prove that their intersection with $L^\infty$ is an algebra for pointwise multiplication. Our results apply to nilpotent Lie groups and Grushin settings.
We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.
We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtained result correctly reproduces the first non-trivial heat kernel coefficient as well as the exact heat kernel diagonals on two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal af…
Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying Hörmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator $$ \mathcal{H}:=\sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-\partial_{t}% $$ where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}…
This paper deals with the randomized heat equation defined on a general bounded interval $[L_1,L_2]$ and with non-homogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on $[0,1]$ with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on $[0,1]$ with homogeneous boundary conditions can be approximated. Via the changes of variable …
We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing $ L^{\infty}_{x}(\mathbb{R}^{n})$ by $BMO_{x}(\mathbb{R}^{n})$ and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the Strichartz estimates for the fractional heat equations with a time dependent potential of an…
Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities
We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.
In this paper, we consider a linear heat equation with constant coefficients and a single constant delay. Such equations are commonly used to model and study various problems arising in ecology and population biology when describing the temporal evolution of human or animal populations accounting for migration, interaction with the environment and certain aftereffects caused by diseases or enviromental polution, etc. Whereas dynamical systems with lumped parameters have been addressed in numerous investigations, there still remain a lot of open questions for the case of systems with distribute…
In this work, we study the Cauchy problem of the spatially homogeneous Landau equation with hard potentials in a close-to-quilibrium framework. We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an L2 initial datum for positive time. So that the smoothing effect of Cauchy problem for the spatially homogeneous Landau equation with hard potentials is exactly same as heat equation.
This paper is devoted to the homogenization of the heat conduction equation, with a homogeneous Dirichlet boundary condition, having a periodically oscillating thermal conductivity and a vanishing volumetric heat capacity. A homogenization result is established by using the evolution settings of multiscale and very weak multiscale convergence. In particular, we investigate how the relation between the volumetric heat capacity and the microscopic structure effects the homogenized problem and its associated local problem. It turns out that the properties of the microscopic geometry of the proble…
We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.
In this work, we study the nonlinear spatially homogeneous Landau equation with hard potential in a close-to-equilibrium framework, we show that the solution to the Cauchy problem with $L^2$ initial datum enjoys a analytic Gelfand-Shilov regularizing effect in the class $S^1_1(\mathbb{R}^3)$, meaning that the solution of the Cauchy problem and its Fourier transformation are analytic for any positive time, the evolution of analytic radius is similar to the heat equation.
We consider the initial boundary value problem of non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random perturbation. The space boundary is Lipschitz and we impose non-zero cylinder condition. We prove a regularity result after finding suitable spaces for the solution and the pre-assigned datum in the problem. The tools from potential theory, harmonic analysis and probability are used. Some Lemmas are as important as the main Theorem.
We study the reduction of the heat equation in Riemannian spaces which admit a gradient Killing vector, a gradient homothetic vector and in Petrov Type D,N,II and Type III space-times. In each reduction we identify the source of the Type II hidden symmetries. More specifically we find that a) If we reduce the heat equation by the symmetries generated by the gradient KV the reduced equation is a linear heat equation in the nondecomposable space. b) If we reduce the heat equation via the symmetries generated by the gradient HV the reduced equation is a Laplace equation for an appropriate metric.…
This is the second part of a two parts work on the analysis of heat-type equations on manifolds with fibered boundary equipped with a $Φ$-metric. This setting generalizes the asymptotically conical (scattering) spaces and includes special cases of magnetic and gravitational monopoles. The core of this second part consists on the construction of parametrix for heat-type equations. Consequently we use the constructed parametrix to infer results regarding existence and regularity of certain homogeneous and non homogeneous second order linear parabolic equations with non constant coefficients. Thi…
We consider a model of chemically reacting heat conducting compressible mixture. We investigate the corresponding system of partial differential equations in the steady regime with slip boundary conditions for the velocity and, in dependence on the model parameters, we establish existence of either weak or variational entropy solutions. The results extend the range of parameters for which the existence of weak solutions is known in the case of homogeneous Dirichlet boundary conditions for the velocity.
Four problems about recovery of a high-frequency source in the one-dimension heat equation with homogeneous initial-boundary conditions by some information about partial asymptotic of its solution have solved. It is shown, that the source can be completely recovered from a specific data about incomplete (two-terms) asymptotic of the solution. Before formulation of each problem about recovery of a source, construction and justification of the asymptotic of the solution of original initial-boundary problem is given.
In this work we study a quasi-static evolution of thermo-visco-elastic model with homogeneous thermal expansion. We assume that material is subject to two kinds of mechanical deformations: elastic and inelastic. Inelastic deformation is related to a hardening rule of Norton-Hoff type. Appearance of inelastic deformation causes transformation of mechanical energy into thermal one, hence we also take into the consideration changes of material's temperature. The novelty of this paper is to take into account the thermal expansion of material. We are proposing linearisation of the model for homogen…
This paper is devoted to global well-posedness, self-similarity and symmetries of solutions for a superdiffusive heat equation with superlinear and gradient nonlinear terms with initial data in new homogeneous Besov-Morrey type spaces. Unlike the heat equation, we need to develop an appropriate decomposition of the two-parametric Mittag-Leffler function in order to obtain Mikhlin-type estimates get our well-posedness theorem. To the best of our knowledge, the present work is the first one concerned with a well-posedness theory for a time-fractional partial differential equations of order $α\in…
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.
The goal of this article is to show a local exact controllability to smooth (C2) trajectories for the 2-d density dependent incompressible Navier-Stokes equations. Our controllability result requires some geometric condition on the ow of the target trajectory, which is remanent from the transport equation satisfied by the density. The proof of this result uses a fixed point argument in suitable spaces adapted to a Carleman weight function that follows the ow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations.
In this work, we present new results on solvability of the equation $A^{*}(D)f=μ$ for $f \in L^{p}$ and positive measure data $μ$ associated to an elliptic homogeneous linear differential operator $A(D)$ of order m. Our method is based on $(m,p)-$energy control of $μ$ giving a natural characterization for solutions when $1\leq p < \infty$. We also obtain sufficient conditions in the limiting case $p=\infty$ using {new $L^{1}$ estimates on measures for elliptic and canceling operators.
We examine the Navier-Stokes equations with homogeneous slip boundary conditions coupled with the heat equation with homogeneous Neumann conditions in a bounded domain in $R^3$. The considered domain is a cylinder with $x_3$-axis. The aim of this paper is to show long time estimates without smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of $L_2$ norms of derivatives with respect to $x_3$ of the initial velocity, the initial temperature and the external force.
We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.
The well known heat equation with finite speed of propagation proposed by Cattaneo is obtained by a more general fourth order PDE when a certain (small) parameters is put equal to zero. It seems that this fourth order equation has been essentially overlooked in the literature, in particular when it is subject to non homogeneous boundary conditions. In this paper we examine its well posedness and the asymptotic behavior when certain coefficients tend to singular values.
Hyperbolic systems in one dimensional space are frequently used in modeling of many physical systems. In our recent works, we introduced time independent feedbacks leading to the finite stabilization for the optimal time of homogeneous linear and quasilinear hyperbolic systems. In this work, we present Lyapunov's functions for these feedbacks and use estimates for Lyapunov's functions to rediscover the finite stabilization results.
There is studied problem on solvability of linear non-homogeneous differential equation of higher even order. There is proved the theorem on necessary and sufficient conditions on existence of solutions to the equation in the Schwartz space.
As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm|<C/t curvature decay, we examine the linearization of an equivalent flow at fixed points discovered recently by Baird--Danielo and Lott: nongradient homogeneous expanding Ricci solitons on nilpotent or solvable Lie groups. For all explicitly known nonproduct examples, we demonstrate linear stability of the flow at these fixed points and prove that the linearizations generate strongly continuous semigroups.
The asymptotic behavior of the solutions of the second order linearized Vlasov-Poisson system around homogeneous equilibria is derived. It provides a fine description of some nonlinear and multidimensional phenomena such as the existence of Best frequencies. Numerical results for the 1Dx1D and 2Dx2D Vlasov-Poisson system illustrate the effectiveness of this approach.
