The asymptotic behavior of some semilinear parabolic PDEs is analyzed by means of a "mean value" property. This property allows us to determine, by means of appropriate {\em{a priori}} estimates, some exponential decay results for suitable global solutions. We also apply the method to investigate a well-known finite time blow-up result. An application is given to a one-dimensional semilinear parabolic PDE with boundary degeneracy. Our results shed further light onto the problem of determining initial data for which the corresponding solution is guaranteed to exponentially decay to zero or blow-up in finite time.
We obtain pointwise estimates for solutions of semilinear parabolic equations with a potential on connected domains both of $\mathbb R^n$ and of general Riemannian manifolds.
In this paper, we establish blow-up rates for higher-order semilinear parabolic equations with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions.
In this paper, we establish the existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces for a large class of initial data including distributions other than Radon measures. We also obtain sufficient conditions for the existence of solutions to viscous Hamilton-Jacobi equations.
In this paper we consider an initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal Neumann boundary condition. We prove comparison principle, the existence theorem of a local solution and study the problem of uniqueness and nonuniqueness.
We are concerned with global existence for semilinear parabolic equations on Riemannian manifolds with negative sectional curvatures. A particular attention is paid to the class of initial conditions which ensure existence of global solutions. Indeed, we show that such a class is crucially related to the curvature bounds.
We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as $t\to\infty$.
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.
The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.
We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.
We study the uniqueness of singular radial (forward and backward) self-similar positive solutions of the equation $u_t-Δu = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ where $p\geq(n+2)/(n-2)_+$.
We exhibit a singularly perturbed parabolic problems for which the asymptotic behavior can be described by an one-dimensional ordinary differential equation. We estimate the continuity of attractors in the Hausdorff metric by rate of convergence of resolvent operator.
In this paper, we present output feedback boundary stabilization for a class of semilinear parabolic PDEs with a boundary measurement and an actuation located at the same place. The method uses backstepping transformations, where the state and error systems are proved to be locally exponentially stable in the $\mathbb{H}^4$ norm. The stability of the transformed systems are obtained by constructing a strict Lyapunov function. A numerical example using the FitzHugh-Nagumo equation shows the proposed control law stabilizes the system into its equilibrium solution.
We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete solution sets.
If $p>1+2/n$ then the equation $u_t-Δu = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions lying on the borderline between global existence and blow-up.
In this article, we consider a semilinear pseudo parabolic heat equation with the nonlinearity which is the product of logarithmic and polynomial functions. Here we prove the global existence of solution to the problem for arbitrary dimension $n \geq 1$ and power index $p>1$. Asymptotic behaviour of the solution has been addressed at different energy levels. Moreover, we prove that the global solution indeed decays with an exponential rate. Finally, sufficient conditions are provided under which blow up of solutions take place.
We investigate the large-time behavior of the sign-changing solution of the inhomogeneous semilinear heat equation with a forcing term depending of time and space. we identify the critical exponent for this problem, which separates the nonexistence/existence of global-in-time solutions, and show the discontinuity of this critical exponent at the point which the inhomogeneous term becomes independent of time variable.
We introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain, under boundary condition of the oblique derivative type. We employ these notions in the stability analysis of semilinear problems. Some of the properties we derive are new even in the Dirichlet or in the whole space cases. As an application, we show the validity of the hair-trigger effect for the Fisher-KPP equation on general, uniformly smooth domains.
In this work, we study the stable determination of four space-dependent coefficients appearing in a coupled semilinear parabolic system with variable diffusion matrices subject to dynamic boundary conditions which couple intern-boundary phenomena. We prove a Lipschitz stability result for interior and boundary potentials by means of only one observation component, localized in any arbitrary open subset of the physical domain. The proof mainly relies on some new Carleman estimates for dynamic boundary conditions of surface diffusion type.
Let $G=(V,E)$ be a locally finite connected weighted graph, and $Ω$ be an unbounded subset of $V$. Using Rothe's method, we study the existence of solutions for the semilinear heat equation $\partial_tu+|u|^{p-1}\cdot u=Δu~(p\ge1)$ and the parabolic variational inequality \begin{eqnarray*} \int_{Ω^\circ} \partial_tu\cdot(v-u)\,dμ\ge \int_{Ω^\circ}(Δu+f)\cdot(v-u)\,dμ\qquad\mbox{for any }v\in \mathcal{H}, \end{eqnarray*} where $\mathcal{H}=\{u\in W^{1,2}(V):u=0\mbox{ on }V\backslashΩ^\circ\}$.
We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations.
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 the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the Dirichlet-to-Neumann map measured at a single boundary point, or integrated against a fixed measure. This result is valid even when the Dirichlet data is only given on a small subset of the boundary. We also give related uniqueness results on Riemannian manifolds.
The objective of this article is to study the boundary value problem for the general semilinear elliptic equation of second order involving $L^1$ functions or Radon measures with finite total variation. The study investigates the existence and uniqueness of `{\it very weak}' solutions to the boundary value problem for a given $L^1$ function. However, a `{\it very weak}' solution need not exist when an $L^1$ function is replaced with a measure due to which the corresponding reduced limits has been found for which the problem admits a solution in a `{\it very weak}' sense.
The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,\varepsilon}$-regularity for such manifolds (for some positive, but small $\varepsilon$). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $n\in\mathbb N$ is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the …
Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
In this paper, we study inverse boundary problems associated with semilinear parabolic systems in several scenarios where both the nonlinearities and the initial data can be unknown. We establish several simultaneous recovery results showing that the passive or active boundary Dirichlet-to-Neumann operators can uniquely recover both of the unknowns, even stably in a certain case. It turns out that the nonlinearities play a critical role in deriving these recovery results. If the nonlinear term belongs to a general $C^1$ class but fulfilling a certain growth condition, the recovery results are …
We investigate nodal radial solutions to semilinear problems of type \[\begin{cases}-Δu = f(|x|,u) \qquad & \text{ in } Ω, \newline u= 0 & \text{ on } \partial Ω, \end{cases} \] where $Ω$ is a bounded radially symmetric domain of $\mathbb R^N$ ($N\ge 2$) and $f$ is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse index in a forthcoming work…
