We continue our study of bounded solutions of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ Assuming that the initial value $u_0=u(\cdot,0)$ of the solution has finite limits $θ^\pm$ as $x\to\pm\infty$, our goal is to describe the asymptotic behavior of $u(x,t)$ as $t\to\infty$. In a prior work, we showed that if the two limits are distinct, then the solution is quasiconvergent, that is, all its locally uniform limit profiles as $t\to\infty$ are steady states. It is known that this result is not valid in general if the limits are equal: $θ^\pm=θ_0$. In the present paper, we have a closer look at the equal-limits case. Under minor non-degeneracy assumptions on the nonlinearity, we show that the solution is quasiconvergent if either $f(θ_0)\ne0$, or $f(θ_0)=0$ and $θ_0$ is a stable equilibrium of the equation $\dot ξ=f(ξ)$. If $f(θ_0)=0$ and $θ_0$ is an unstable equilibrium of the equation $\dot ξ=f(ξ)$, we also prove some quasiconvergence theorem making (necessarily) additional assumptions on $u_0$. A major ingredient of our proofs of the quasiconvergence theorems---and a result of independent interest---is the classification of entire solutions of a certain type as steady states and heteroclinic connections between two disjoint sets of steady states.
We consider the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ We prove that if a solution $u$ of this equation is bounded and its initial value $u(x,0)$ has distinct limits at $x=\pm\infty,$ then the solution is quasiconvergent, that is, all its limit profiles as $t\to\infty$ are steady states.
We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.
We consider solutions to linear parabolic equations with initial data decaying at spatial infinity. For a class of advection-diffusion equations with a spatially dependent velocity field, we study the behavior of solutions as time tends to infinity. We characterize velocity fields, so that positive solutions decay or lift-off at spatial infinity as time tends to infinity. This addresses the question of stability of the zero solution for decaying perturbations.
This paper is concerned with the bifurcation from infinity of the nonlinear Schrödinger equation $$-Δu+V(x)u=λu+f(x,u),\hspace{0.4cm} x\in \mathbb{R}^N.$$ We treat this problem in the framework of dynamical systems by considering the corresponding parabolic equation on unbounded domains. Firstly, we establish a global invariant manifold for the parabolic equation on $\mathbb{R}^N$. Then, we restrict the parabolic equation to this invariant manifold, which generates a system of finite dimension. Finally, we use the Conley index theory and the shape theory of attractors to establish some new res…
We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem $u_t = Δ_\infty^h u$ where $Δ_\infty^h$ is the $h$-homogeneous operator associated with the infinity-Laplacian, $Δ_\infty^h u = |Du|^{h-3} < D^2uDu,Du>$. We also derive the asymptotic behavior of $u$ for the problem posed in the whole space and for the Dirichlet problem with zero boundary conditions.
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.
This paper is devoted to the generic transversality of radially symmetric stationary solutions of nonlinear parabolic systems of the form \[ \partial_t w(x,t) = -\nabla V\bigl(w((x,t))\bigr) + Δ_x w(x,t) \,, \] where the space variable $x$ is multidimensional and unbounded. It is proved that, generically with respect to the potential $V$, radially symmetric stationary solutions that are stable at infinity (in other words, that approach a minimum point of $V$ at infinity in space) are transverse; as a consequence, the set of such solutions is discrete. This result can be viewed as the extension…
This is a continuation, and conclusion, of our study of bounded solutions $u$ of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line whose initial data $u_0=u(\cdot,0)$ have finite limits $θ^\pm$ as $x\to\pm\infty$. We assume that $f$ is a locally Lipschitz function on $\mathbb{R}$ satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of $u(x,t)$ as $t\to\infty$. In the first two parts of this series we mainly considered the cases where either $θ^-\neq θ^+$; or $θ^\pm=θ_0$ and $f(θ_0)\ne0$; or else $θ^\pm=θ_0$, $f(θ_0)=0$, and $θ_0$ is a st…
We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the value…
We consider degenerate Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative. It is well known that these equations admit global solutions when epsilon is small enough, and that these solutions decay as t -> +infinity with the same rate of solutions of the limit problem (of parabolic type). In this paper we prove decay-error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem. These estimates show in the same time that the difference tends to zero both as epsilon -> 0, and as…
We investigate the issue of uniqueness of the limit flow for a relevant class of quasi-linear parabolic equations defined on the whole space. More precisely, we shall investigate conditions which guarantee that the global solutions decay at infinity uniformly in time and their entire trajectory approaches a single steady state as time goes to infinity. Finally, we obtain a characterization of solutions which blow-up, vanish or converge to a stationary state for initial data of the form $λφ_0$ while $λ>0$ crosses a bifurcation value $λ_0$.
This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)ΔU(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive. Under the behavior of the diffusion coefficient $D$ with polynomial order at spatial infinity, a family of supersolutions with slowly decaying property at spatial infinity is provided. As a first application, weighted $L^2$ type decay estimates for the initial-boundary value problem of the corresponding parabolic equation are proved. The second application is the …
In this paper there are estimated the derivatives of the solution of an initial boundary value problem for a nonlinear uniformly parabolic equation in the interior with the total variation of the boundary data and the L^{infinity}-norm of the initial condition.
We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t tends to +infinity. The case where b(t) behaves like (1+t)^{-p} with p<1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t tends to +infinity and as epsilon…
In this paper, we study the numerical approximation of a coupled system of elliptic-parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an unsaturated heterogeneous medium with distributed microstructures as they often arise in modeling reactive flow in cementitious-based materials. Besides ensuring the well-posedness of our two-scale model, we design two-scale convergent numerical approximations and prove a priori error estimates for the semidiscrete case. We complement our analysis with simulat…
Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of an isolated invariant set. As the frequency of the oscillations tends to infinity, we prove that every isolated invariant set of the averaged autonomous equation can be continued to an isolated invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index …
Criteria for the existence of $T$-periodic solutions of nonautonomous parabolic equation $u_t = Δu + f(t,x,u)$, $x\in\mathbb{R}^N$, $t>0$ with asymptotically linear $f$ will be provided. It is expressed in terms of time average function $\hat f$ of the nonlinear term $f$ and the spectrum of the Laplace operator $Δ$ on $\mathbb{R}^N$. One of them says that if the derivative $\hat f_\infty$ of $\hat f$ at infinity does not interact with the spectrum of $Δ$, i.e. $\mathrm{Ker} (-Δ+ \hat f_\infty)=\{0\}$, then the parabolic equation admits a $T$-periodic solution. Another theorem is derived in the…
Under a precise nonlinearity-diffusivity assumption we establish the decay of entropy solutions of a degenerate nonlinear parabolic equation with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure).
We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, we obtain the Liouville-type theorem in the case when the coefficients may grow at the spacial infinity. We also study a corresponding parabolic problem in the $n$-dimensional whole space and discuss the energy identity for solutions in $L^q$. As a co…
We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behavi…
We consider a family of Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipation term with a coefficient which tends to 0 as t -> +infinity. It is well-known that, when the decay of the coefficient is slow enough, solutions behave as solutions of the corresponding parabolic equation, and in particular they decay to 0 as t -> +infinity. In this paper we consider the nondegenerate and coercive case, and we prove optimal decay estimates for the hyperbolic problem, and optimal decay-error estimates for the difference between solutions of th…
This paper is concerned with the asymptotic behavior of the solution to the semilinear parabolic equation with dynamical boundary condition. Our main goal is to prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz--Simon type inequality.
By employing a Carnot parabolic maximum principle, we show existence-uniqueness of viscosity solutions to a class of equations modeled on the parabolic infinite Laplace equation in Carnot groups. We show stability of solutions within the class and examine the limit as t goes to infinity.
We consider a class of second order linear nonautonomous parabolic equations in R^d with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator G(t,s) be compact in C_b(R^d) for t>s, and describe the asymptotic behavior of G(t,s)f as t-s goes to infinity in terms of a family of measures mu_s, s in R, solution of the associated Fokker-Planck equation.
We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method provided by E.E. Levi.
We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t tends to +infinity. We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t tends to +infinity, with the same rate of the solution of the limit problem of parabolic type.
In this article, we investigate the initial and boundary blow-up problem for the $p$-Laplacian parabolic equation $u_t-Δ_p u=-b(x,t)f(u)$ over a smooth bounded domain $Ω$ of $\mathbb{R}^N$ with $N\ge2$, where $Δ_pu={\rm div}(|\nabla u|^{p-2}\nabla u)$ with $p>1$, and $f(u)$ is a function of regular variation at infinity. We study the existence and uniqueness of positive solutions, and their asymptotic behaviors near the parabolic boundary.
We prove in this short report the existence of a fundamental solution (F.S.) for the Cauchy initial boundary problem on the whole space for the parabolic differential equation having at origin the point of non-integrable unbounded discontinuity for coefficient before a first order derivative. We give also the non-asymptotic rapidly decreasing at infinity estimate for these function. We extend the classical parametrix method offered by E.E.Levi.
The fractional heat operator $(\partial_t-Δ_x)^s$ and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to $(\partial_t-Δ_x)^s$ and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator.
If $Ω$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-Δu+f(u)=0$ in $Q_\infty^Ω:=Ω\times (0,\infty)$, $u=\infty$ on the parabolic boundary $\partial_{p}Q$. We prove that in most cases, the existence and uniqueness is reduced to the same property for the associated stationary equation in $Ω$.
We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous results obtained assuming a linear growth.
We prove existence of the largest and the smallest entropy solutions to the Cauchy problem for a nonlinear degenerate anisotropic parabolic equation. Applying this result, we establish the comparison principle in the case when at least one of the initial functions is periodic. In the case when initial function vanishes at infinity (in the sense of strong average) we prove the long time decay of an entropy solution under exact nonlinearity-diffusivity condition.
This paper is concerned with a general class of fully nonlinear parabolic equations with monotone nonlocal terms. We investigate the quasiconvexity preserving property of positive, spatially coercive viscosity solutions. We prove that if the initial value is quasiconvex, the viscosity solution to the Cauchy problem stays quasiconvex in space for all time. Our proof can be regarded as a limit version of that for power convexity preservation as the exponent tends to infinity. We also present several concrete examples to show applications of our result.
In this paper, we are concerned with the instability and stability of a quasi-linear hyperbolic-parabolic system modeling vascular networks. Under the assumption that the pressure satisfies $\frac{νP'(\barρ)}{γ\barρ} < β$, we first show that the steady-state is linear unstable (i.e., the linear solution grows in time in $L^2$) by constructing an unstable solution. Then based on the lower grow estimates on the solution to the linear system, we prove that the steady-state is nonlinear unstable in the sense of Hadamard. On the contrary, if the pressure satisfies $\frac{νP'(\barρ)}{γ\barρ} > β$, w…
We investigate the effects of advection on the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions. Various asymptotic behaviors of the principal eigenvalues, when advection coefficient approaches infinity, are established in heterogeneous environments, where spatial or temporal degeneracy could occur in the advection term. Our findings partially extend the existing results in Chen-Lou [2008 Indiana Univ. Math. J.] and Peng-Zhou [2018 Indiana Univ. Math. J.] for elliptic operators and those in Peng-Zhao [2015 Calc. Var. Partial Diff.] for par…
In this paper, we derive a priori estimates for the gradient and second order derivatives of solutions to a class of Hessian type fully nonlinear parabolic equations with the first initial-boundary value problem on Riemannian manifolds. These a priori estimates are derived under conditions which are nearly optimal. Especially, there are no geometric restrictions on the boundary of the Riemannian manifolds. And as an application, the existence of smooth solutions to the first initial-boundary value problem even for infinity time is obtained.
Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and the nonlinear absorption. In particular, the limit as time goes to infinity of the mass of integrable solutions is identified, together with the rate of expansion of the support for compactly supported initial data. The persistence of dead cores is also shown. The proof of these results strongly relies on gradient estimates which are first established.
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 consider a non-local diffusion equation involving the fractional $p(x)$-Laplacian with nonlinearities of variable exponent type. Employing the sub-differential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.
In this paper, we discuss singular Neumann boundary problem for a class of nonlinear parabolic equations in one space dimension. Our boundary problem describes motion of a planar curve sliding along the boundary with a zero contact angle, which can be viewed as a limiting model for the capillary phenomenon. We study the uniqueness and existence of solutions by using the viscosity solution theory. We also show the convergence of the solution to a traveling wave as time proceeds to infinity when the initial value is assumed to be convex.
In this paper, we consider a nonlocal parabolic equation associated with initial and Dirichlet boundary conditions. Firstly, we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells. Then we obtain a threshold of global existence and blow up for solutions with critical initial energy. Furthermore, for those solutions satisfy $J(u_0)\leq d$ and $I(u_0)\neq 0$, we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow-up solutions increase exponentially.
We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast.
We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Control problems play a key role in engineering, economics and sciences. To be more precise, in climate sciences, often times, relevant problems are formulated in long time scales, so that, the problem of possible asymptotic behaviors when the time-horizon goes to infinity becomes natural. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary …
