In this paper, we consider the global Cauchy problem for the $L^2$-critical semilinear heat equations $
\partial_t h=Δh\pm |h|^{\frac4d}h, $
with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with $s>0$. {\it First,} we prove that there exists some positive constant $γ_0$ depending on $d$, such that the Cauchy problem is locally and globally well-posed for any initial data $h_0$ which is radial, supported away from the origin and in the negative Sobolev space $\dot H^{-γ_0}(\R^d)$. In particular, it leads to local and global existences of the solutions to Cauchy problem considered above for the initial data in a proper subspace of $L^p(\R^d)$ with some $p<2$. {\it Secondly,} the sharp asymptotic behavior of the solutions ( i.e. $L^2$-decay estimates ) as $t\to +\infty$ are obtained with arbitrary large initial data $h_0\in \dot H^{-γ_0}(\R^d)$ in the defocusing case and in the focusing case with suitably small initial data $h_0$.
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.
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 obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.
Existence of strong solutions to a nonlocal semilinear heat equation is shown. The main feature of the equation is that the nonlocal term depends on the unknown on the whole time interval of existence, the latter being given a priori. The proof relies on Schauder's fixed point theorem and semigroup theory.
We study the Cauchy problem for a semilinear heat equation with initial data non-rarefied at $\infty$. Our interest lies in the discussion of the effect of the non-rarefied factors on the life span of solutions, and some sharp estimates on the life span is established.
Existence of global solutions to initial value problems for a discrete analogue of a d-dimensional semilinear heat equation is investigated. We prove that a parameter αin the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for 0 < α\le 2/d, and, for α> 2/d, existence of non-trivial global solutions for sufficiently small initial data.
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 consider semilinear parabolic equations with nonlinear boundary conditions. We give conditions which guarantee global existence of solutions as well as blow-up in finite time of all solutions with nontrivial initial data. The results depend on the behavior of variable coefficients as $t \to \infty.$
We study the initial trace problem for positive solutions of semilinear heat equations with strong absorption. We show that in general this initial trace is an outer regular Borel measure. We emphasize in particular the case where $u$ satisfies (E) $\partial_t u-Δu+t^α|u|^{q-1}u=0$, with $q>1$ and $α>-1$ and prove that in the subcritical case $1
We consider a semilinear parabolic equation with flux at the boundary governed by a nonlinear memory. We give some conditions for this problem which guarantee global existence of solutions as well as blow up in finite time of all nontrivial solutions. The results depend on the behavior of variable coefficients as $t \to \infty.$
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 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 study the Cauchy problem for the semilinear fractional heat equation $u_{t}=\triangle^{α/2}u+f(u)$ with non-negative initial value $u_{0}\in L^{q}(\mathbb{R}^{n})$ and locally Lipschitz, non-negative source term $f$. For $f$ satisfying the Osgood-type condition $\int_{1}^{\infty}\frac{ds}{f(s)}=\infty$, we show that there exist initial conditions such that the equation has no local solution in $L^{1}_{loc}(\mathbb{R}^{n})$.
We give a simple proof of a lower bound for the Dirichlet heat kernel in terms of the Gaussian heat kernel. Using this we establish a non-existence result for semilinear heat equations with zero Dirichlet boundary conditions and initial data in $L^q(Ω)$ when the source term $f$ is non-decreasing and $\limsup_{s\to\infty}s^{-γ}f(s)=\infty$ for some $γ>q(1+2/n)$. This allows us to construct a locally Lipschitz $f$ satisfying the Osgood condition $\int_{1}^{\infty}1/f(s)\ \,\d s =\infty$, which ensures global existence for bounded initial data, such that for every $q$ with $1\le q<\infty$ there i…
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 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.
The purpose in this paper is to determine the global behavior of solutions to the initial-boundary value problems for energy-subcritical and critical semilinear heat equations by initial data with lower energy than the mountain pass level in energy spaces associated with self-adjoint operators satisfying Gaussian upper bounds. Our self-adjoint operators include the Dirichlet Laplacian on an open set, Robin Laplacian on an exterior domain, and Schrödinger operators, etc.
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.
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.
Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat Friedmann-Lemaître-Robertson-Walker spacetimes. We treat the so-called heatlike case where the critical exponent is affected by the Fujita exponent. We show upper bounds of the lifespan of blow-up solutions by distinguishing subcritical and critical cases.
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 $\mathcal{H}=\{u\in W^{1,2}(V):u=0\mbox{ on }V\backslashΩ^\circ\}$.
In this paper we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $ f ( u ) = λ|u|^αu$. We show that low regularity of $f$ (i.e., $α>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_t= f(w)$. This yields in particular an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields…
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 paper, we study a critical exponent to the semilinear heat equation with forcing term on Heisenberg group. Our technique of proof is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg group. Surprisingly, the critical exponent will be bounded for all dimensions of the Heisenberg group, in contrast to the Euclidean case which in the 1D and 2D cases will be infinite. In addition, we give an upper bound estimate on the lifespan of local solutions.
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.
The local and global existence of the Cauchy problem for semilinear heat equations with small data is studied in the weighted $L^\infty (\mathbb R^n)$ framework by a simple contraction argument. The contraction argument is based on a weighted uniform control of solutions related with the free solutions and the first iterations for the initial data of negative power.
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.
We construct a periodic solution to the semilinear heat equation with power nonlinearity, in one space dimension, 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. Thanks to the geometrical interpretation of the finite-dimensional parameters in terms of the blow-up time and blow-up point, we derive the stability of the constructed solution with respect to initial data.
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…
The problem of obtaining necessary and sufficient conditions for local existence of non-negative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term $f$ has only recently been resolved (Laister et al. (2016)). There, for the more difficult case of initial data in $L^1$, a necessary and sufficient integral condition on $f$ emerged. Here, subject to this integral condition, we consider other fundamental properties of solutions with $L^1$ initial data of indefinite sign, namely: uniqueness, regularity, continuous dependence and comparison. We als…
We consider a linear nonlocal heat equation in a bounded domain $Ω\subset\mathbb{R}^d$ with Dirichlet boundary conditions. The non-locality is given by the presence of an integral kernel. We analyze the problem of controllability when the control acts on an open subset of the domain. It is by now known that the system is null-controllable when the kernel is time-independent and analytic or, in the one-dimensional case, in separated variables. In this paper, we relax this assumption and we extend the result to a more general class of kernels. Moreover, we get explicit estimates on the cost of n…
A semilinear heat equation $u_{t}=Δu+f(u)$ with nonnegative initial data in a subset of $L^{1}(Ω)$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $Ω\subseteq \R^{n}$. A simple technique for proving existence and regularity based on the existence of supersolutions is presented, then a method of construction of local and global supersolutions is proposed. This approach is applied to the model case $f(s)=s^{p}$, $ϕ\in L^{q}(Ω)$: new sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical range of…
We consider the homogenization of a semilinear heat equation with vanishing viscosity and with oscillating positive potential depending on $u/\varepsilon$. According to the rate between the frequency of oscillations in the potential and the vanishing factor in the viscosity, we obtain different regimes in the limit evolution and we discuss the locally uniform convergence of the solutions to the effective problem. The interesting feature of the model is that in the strong diffusion regime the effective operator is discontinuous in the gradient entry. We get a complete characterization of the li…
Let $G=(V,E)$ be a finite or locally finite connected weighted graph, $Δ$ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on $G$ \begin{equation*} \left\{ \begin{array}{lc} u_t=Δu + u^{1+α} &\, \text{in $(0,+\infty)\times V$,}\\ u(0,x)=a(x) &\, \text{in $V$.} \end{array} \right. \end{equation*} We conclude that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ and $V(x,r)\simeq r^m$, if $0<mα<2$, then the non-negative solution $u$ is not global, and if $mα>2$, then t…
We consider the semilinear Lane Emden problem in a smooth bounded simply connected domain in the plane, invariant by the action of a finite symmetry group G. We show that if the orbit of each point in the domain, under the action of the group G, has cardinality greater than or equal to four then, for p sufficiently large, there exists a sign changing solution of the problem with two nodal regions whose nodal line does not touch the boundary of the domain. This result is proved as a consequence of an analogous result for the associated parabolic problem.
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…
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…
We study the $L^{p}$-solutions for the semilinear heat equation with unbounded coefficients and driven by a infinite dimensional fractional Brownian motion with self-similarity parameter $H > 1/2$. Existence and uniqueness of local mild solutions are showed.
We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We introduce a new supersolution which plays a crucial role. Our method does not rely on a change of variables, and hence it can be applied to a wide class of nonlocal parabolic equations. In particular, when the nonlinear term is $u^p$ or $e^u$, a local-in-time solution can be constructed in the critic…
In this paper, we consider the semilinear heat equations under Dirichlet boundary condition \[ u_{t}\left(x,t\right)=Δu\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in Ω\times\left(0,+\infty\right), u\left(x,t\right)=0, & \left(x,t\right)\in\partial Ω\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0, & x\in\overlineΩ, \] where $Ω$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partialΩ$. The main contribution of our work is to introduce a new condition \[ (C) α\int_{0}^{u}f(s)ds \leq uf(u)+βu^{2}+γ,\,\,u>0 \] for some $α, β, γ>0$ with $0<β\leq\frac{\left(α-…
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.
Here we design boundary feedback stabilizers to unbounded trajectories, for semi-linear stochastic heat equation with cubic non-linearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. Then, the simple form of the feedback, we propose here, allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed…
We will present some rigidity results for solutions to semilinear elliptic equations of the form $\Deltau = W'(u)$, where W is a quite general potential with a local minimum and a local maximum. We are particularly interested in Liouvlle-type theorems and symmetry results, which generalise some known facts about the Cahn-Hilliard equation.
