In the paper we prove the existence of probabilistic solutions to systems of the form $-Au=F(x,u)+μ$, where $F$ satisfies a generalized sign condition and $μ$ is a smooth measure. As for $A$ we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on $L^1$. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer's property (L) of Markov processes and in terms of order compactness of the associated resolvent.
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-…
We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in $% \mathbb{R} ^{n}$, then the semigroup generated by the considered problem possesses a global attractor in $H^{2}\left( \mathbb{R} ^{n}\right) \times L^{2}\left( \mathbb{R} ^{n}\right) $. We also establish boundedness of this attractor in $ H^{3}\left( \mathbb{R} …
The purpose of this paper is to introduce a new condition \[ \hbox{(C)$\hspace{1cm} α\int_{0}^{u}f(s)ds \leq uf(u)+βu^{2}+γ,\,\,u>0$} \] for some $α, β, γ>0$ with $0<β\leq\frac{\left(α-2\right)λ_{0}}{2}$, where $λ_{0}$ is the first eigenvalue of discrete Laplacian $Δ_ω$, with which we obtain blow-up solutions to discrete semilinear heat equations \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=Δ_ωu\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in S\times\left(0,+\infty\right),\\ u\left(x,t\right)=0, & \left(x,t\right)\in\partial S\times\left[0,+\infty\right),\\ u\left(x,0\right)=u_{0}\ge…
We consider the second order semilinear elliptic system $Δu= p\left( x\right) v^α,$ $Δv= q\left(x\right) u^β,$ where $x \in \mathbf{R}^N,$ $N \geq 3,$ $α$ and $β$ are positive constants, $p$ and $q$ are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions $p$ and $q$ are of slow decay.
We study the semilinear elliptic equation \begin{equation*} -Δu=u^α|\log u|^β\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2} < α< \frac{n+2}{n-2}$ and $-\infty<β<\infty$. Our main result establishes that nonnegative solution $u\in C^2(B_1\setminus\{0\})$ of the above equation either has a removable singularity at the origin or behaves like \begin{equation*} u(x) = A(1+o(1)) |x|^{-\frac{2}{α-1}} \left(\log \frac{1}{|x|}\right)^{-\fracβ{α-1}}\quad\text{as } x\rightarrow 0, \end{equation*} with \begin{equation*} A=\left[\left(\frac{2…
We consider the semilinear damped wave equation $\partial_{tt}^2 u(x,t)+γ(x)\partial_t u(x,t)=Δu(x,t)-αu(x,t)-f(x,u(x,t))$. In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $γ$ does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that $\|e^{At}A^{-1}\|\leq h(t)$ for some function $h$ with $h(t)\rightarrow 0$ when $t\rightarrow +\infty$. We provide general tools to deal with the semilinear stabilization problem in the…
In this paper we prove some symmetry results for entire solutions to the semilinear equation $-Δu=f(u)$, with $f$ nonincreasing in a right neighbourhood of the origin. We consider solutions decaying only in some directions and we give some sufficient conditions for them to be radially symmetric with respect to those variables, such as periodicity or the pointwise decay of some derivatives.
We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We prove also regularity properties and approximation results for solutions of the problem.
In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations % \[ u_{tt}+(-Δ)^δu_{tt}+(-Δ)^αu+(-Δ)^θu_t=|u_t|^p, \quad t\geq 0,\,\, x\in\R^n,\] % with $p>1$, $2θ\in [0, α]$ and $δ\in (θ,α]$. We show that, under additional regularity $\left(H^{α+δ}(\R^n)\cap L^{m}(\R^n) \right)\times \left(H^{2δ}(\R^n)\cap L^{m}(\R^n)\right) $ for initial data, with $m\in (1,2]$, the critical exponent is given by $p_c=1+\frac{2mθ}{n}$. The nonexistence of global solutions in the subcritical cases i…
The question of triviality of solutions of the semilinear Ornstein-Uhlenbeck equation, \[ Δw-\frac{1}{2} \langle x,\nabla w\rangle-\fracλ{p-1}w+|w|^{p-1}w=0, \] is considered. It is shown, that if $p>1$ is Sobolev subcritical or critical and $λ\leq 1$, then all bounded entire solutions are constant. Moreover, in the critical case, the same conclusion holds in the subclass of radial solutions provided that $n\geq 4$ and $λ\in \left[\frac{3 n}{2(n-1)},2\right]$.
We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over R^d and in L^p-spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in IxR^d, (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.
We consider weakly coupled systems of semilinear viscoelastic wave equations with different power source nonlinearities in $\mathbb{R}^n$, $n\geq1$ as follows: \begin{equation*} \left\{\begin{aligned} &u_{tt}-Δu+g\astΔu+u_t=|\partial_t^{\ell}v|^p,\\ &v_{tt}-Δv+g\astΔv+v_t=|\partial_t^{\ell}u|^q,\\ \end{aligned}\right. \end{equation*} with $\ell=0,1$ and $p,q>1$. After presenting $L^m-L^r$ estimates with $1\leq m\leq r\leq \infty$ of solutions to the corresponding linearized problem with vanishing right-hand side, we prove the existence of global in time solutions to the weakly coupled systems,…
In this article, we study the existence of non-trivial weak solutions for the following boundary-value problem \begin{gather*} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=f(x,y,u) \quad\text{ in }Ω, \ u=0 \quad\text{ on }\partialΩ, \end{gather*} where $Ω$ is a bounded domain with smooth boundary in $\mathbb{R}^2, Ω\cap \{x=0\}\ne \emptyset,$ $k >0,$ $f(x,y,0)=0. $
In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -Δu= h(x,u)\quad \ &{\rm in}\ Ω,\\[1.5mm] \phantom{ -Δ} \displaystyle u\ge 0\qquad &{\rm on}\ \partialΩ, \end{array}\right. \end{eqnarray*} where $Ω$ is an exterior domain in $R^N$ with $N\ge 3$, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive s…
We consider inverse problems in space-time $(M, g)$, a $4$-dimensional Lorentzian manifold. For semilinear wave equations $\square_g u + H(x, u) = f$, where $\square_g$ denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map $L: f \rightarrow u|_V$, where $V$ is a neighborhood of a time-like geodesic $μ$, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set where waves can propagate from $μ$ and return back. Moreover, on a given space-time $(M, g)$, the source-to-solution map determines som…
In this work, the Cauchy problem for the semilinear Moore-Gibson-Thompson (MGT) equation with power nonlinearity $|u|^p$ on the right-hand side is studied. Applying $L^2-L^2$ estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow-up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills $1 < p \leqslant p_{\mathrm{Str}}(n)$ for $n \geqslant2$ and $p>1$ for $n=1$. Here the Strauss expon…
In this work we consider the boundary blow-up problem $$ \left\{ \begin{array}{ll} Δu = f(u) & \hbox{in } B\\ \ \ u=+\infty & \hbox{on }\partial B \end{array} \right. $$ where $B$ stands for the unit ball of $\mathbb{R}^N$ and $f$ is a locally Lipschitz function which is positive for large values and verifies the Keller-Osserman condition. Under an additional hypothesis on the asymptotic behavior of $f$ we show that all solutions of the above problem are radially symmetric and radially increasing. Our condition is sharp enough to generalize several results in previous literature.
In the paper we consider elliptic equations of the form $-Au=u^{-γ}\cdotμ$, where $A$ is the operator associated with a regular symmetric Dirichlet form, $μ$ is a positive nontrivial measure and $γ>0$. We prove the existence and uniqueness of solutions of such equations as well as some regularity results. We also study stability of solutions with respect to the convergence of measures on the right-hand side of the equation. For this purpose, we introduce some type of functional convergence of smooth measures, which in fact is equivalent to the quasi-uniform convergence of associated potentials…
We consider the following parabolic system whose nonlinearity has no gradient structure: $$\left\{\begin{array}{ll} \partial_t u = Δu + |v|^{p-1}v, \quad & \partial_t v = μΔv + |u|^{q - 1}u,\\ u(\cdot, 0) = u_0, \quad & v(\cdot, 0) = v_0, \end{array}\right. $$ in the whole space $\mathbb{R}^N$, where $p, q > 1$ and $μ> 0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0, v_0)$ simultaneously in $u$ and $v$ only at one blowup point $a$, according to the following asymptotic dynamics: $$\left\{\begin{array}{c} u(x,t)\sim Γ\…
We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland [20] and Matano [44] states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain $Ω$ further satisfies $Ω$ $\cap$ {|x| $\le$ R} = O(R^2), when R $\rightarrow$ +$\infty$. If the domain is a straight cylinder, an additional natural assumption is needed. The…
Let $G=(V,E)$ be a finite or locally finite connected weighted graph, $Δ$ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on $G$ \begin{equation*} \left\{ \begin{array}{lc} u_t=Δu + u^{1+α} &\, \text{in $(0,+\infty)\times V$,}\\ u(0,x)=a(x) &\, \text{in $V$.} \end{array} \right. \end{equation*} We conclude that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ and $V(x,r)\simeq r^m$, if $0<mα<2$, then the non-negative solution $u$ is not global, and if $mα>2$, then t…
Let $L$ be a second order uniformly elliptic differential operator in a domain $D$ of $\mathbb{R}^{d}$, $ψ:\mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing continuous function and let $ξ,g:D\to\mathbb{R}_+$ be locally bounded Borel measurable functions. Under appropriate conditions, we determine a function $φ$ with values in $]0,1]$ such that for every nonnegative solution to inequality $-Lu+ξψ(u) \geq g$ in $D$ and for every $x\in D$, $$ u(x)\geq p(x)\,φ\left(\frac{G_D(ξψ(p))(x)}{p(x)}\right) $$ where $p=G_Dg$ is the Green function of $g$. The function $φ$ is completely determined by $ψ$ and …
We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u' = Lu + f(u)$ in $L^p(X,m)$ for $p \in [1,\infty)$, where $(X,m)$ is a $σ$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $L^p(X,m)$, and $f$ is a strictly increasing, convex, continuous function on $[0,\infty)$ with $f(0) = 0$ and $\int_1^\infty 1/f < \infty$. Since we make no further assumptions on the behaviour of the diffusion, our main result can be seen as b…
We consider stable solutions of semilinear equations in a very general setting. The equation is set on a Polish topological space endowed with a measure and the linear operator is induced by a carré du champs (equivalently, the equation is set in a diffusion Markov triple). Under suitable curvature dimension conditions, we establish that stable solutions with integrable carré du champs are necessarily constant (weaker conditions characterize the structure of the carré du champs and carré du champ itéré). The proofs are based on a geometric Poincaré formula in this setting. From the general the…
The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=Δu+ψ(t)f(u),\,\,\mbox{ in }Ω\times (0,t^{*}), \] under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case $f(u)=u^{p}$. As a matter of fact, we prove: \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if } &\mbox{the function } f \mbox{ satisfies} &\hspace{20mm}\int_{0}^{\infty}ψ(t)\frac{f…
In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equation with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced by the exponential nonlinearity. In this paper, we consider the following semilinear Neumann elliptic problem with the Trudinger-Moser exponential growth: \begin{equation*}\begin{cases} -dΔu_d+u_d=u_d(e^{u^2_{d}}-1)\ \ \mbox{in}\ Ω,\\ \frac{\partial u_d}{\partialν}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial Ω,\\ \end{cases}\end{equa…
We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an $α$-Hölder continuous drift coefficient, if $α\in (2/3,1)$. The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations, differentiability along subspaces and control theoretic results. We st…
Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation
Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition. Novel results include various asymptotic approximations that are, in combination, valid over the entire space and time interval right up to and including the blow-up time. Preliminary results on continuing a numerical solution beyond the singularity are also presented.
We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-Δ)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$. Our main result establishes an $L^\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \leq n < 2(s+2+\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\in(0,1)$ in dimensions $2 \leq n\leq 6$. It applies to all nonlinearities $f\in C^2$. For such parameters $s$ and $n$, our result leads to the regularity of …
We study a time-fractional semilinear heat equation $$\partial^α_t u -Δu = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$ with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^α$ denotes the Caputo derivative of order $α\in (0,1)$. Since the space $L^1(\mathbb{R}^N)$ is scale critical with $p=1+2/N$, this type of equation is known as a doubly critical problem. It is known that the usual doubly critical equation $\partial_t u-Δu=u^p$ does not have nonnegative global-in-time solutions, while the time-fractional problem does. Moreover, there exists a singular initial …
In this paper, we consider the standard semilinear heat equation \begin{eqnarray*} \partial_t u = Δu + |u|^{p-1}u, \quad p >1. \end{eqnarray*} The determination of the (believed to be) generic blowup profile is well-established in the literature, with the solution blowing up only at one point. Though the blow-up of the gradient of the solution is a direct consequence of the single-point blow-up property and the mean value theorem, there is no determination of the final blowup profile for the gradient in the literature, up to our knowledge. In this paper, we refine the construction technique of…
In this paper we consider semilinear elliptic equations with singularities, whose prototype is the following \begin{equation*} \begin{cases} \displaystyle - div \,A(x) D u = f(x)g(u)+l(x)& \mbox{in} \; Ω,\\ u = 0 & \mbox{on} \; \partial Ω,\\ \end{cases} \end{equation*} where $Ω$ is an open bounded set of $\mathbb{R}^N,\, N\geq 1$, $A\in L^\infty(Ω)^{N\times N}$ is a coercive matrix, $g:[0,+\infty)\rightarrow [0,+\infty]$ is continuous, and $0\leq g(s)\leq {{1}\over{s^γ}}+1$ $\forall s>0$, with $0<γ\leq 1$ and $f,l \in L^r(Ω)$, $r={{2N}\over{N+2}}$ if $N\geq 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, …
In this paper, we consider the following free boundary problem $$ (P)\left\{\begin{array}{ll} Δu = λϕ(x)\Sum_{i=1}^n H(u-μ_i )& \quad \mbox{ in }\ Ω=Ω_2\setminus \overlineΩ_1, \\[0.3cm]u =0 &\quad \mbox{ on } \partial Ω_2, \\[0.3cm]u =M &\quad \mbox{ on } \partial Ω_1. \end{array} \right. $$ The domain $Ω$ is a convex ring where $Ω_1$ and $Ω_2$ are bounded convex domain in $\mathbb{R}^N,$ $N\geq 2,$ $H$ is the Heaviside step function, $λ, M,μ_i$ $(i=1,..,n)$ are a given positive real parameters and $ϕ$ is a given function . We show that under suitable conditions, there exists a solution to pro…
Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $δ(x)=\text{dist}(x,\partial Ω)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$ -Δu- \fracμ{δ^2}u= u^p +τ\quad \text{in } Ω, \quad \quad u=ν\quad\text{on } \partial Ω, $$ where $μ\in \mathbb{R}$, $p>0$, $τ$ and $ν$ are measures on $Ω$ and $\partial Ω$ respectively. We then establish existence results for the system $$ \left\{ \begin{aligned} &-Δu- \fracμ{δ^2}u = ε\, v^p +τ\quad \text{in } Ω, \\ &-Δv- \fracμ{δ^2}v = ε\, u^{\tilde p}+\tilde τ\quad \text{in } Ω…
We consider the following semilinear wave equation with time-dependent damping. \begin{align} \tag{NLDW} \left\{ \begin{array}{ll} \partial_t^2 u - Δu + b(t)\partial_t u = |u|^{p}, & (t,x) \in [0,T) \times \mathbb{R}^n, \\ u(0,x)=\varepsilon u_0(x), u_t(0,x)=\varepsilon u_1(x), & x \in \mathbb{R}^n, \end{array} \right. \end{align} where $n \in \mathbb{N}$, $p>1$, $\varepsilon>0$, and $b(t)\thickapprox (t+1)^{-β}$ with $β\in [-1,1)$. It is known that small data blow-up occurs when $1<p< p_F$ and, on the other hand, small data global existence holds when $p>p_F$, where $p_F:=1+2/n$ is the Fujita…
We analyze multidimensional Markovian integral equations that are formulated with a time-inhomogeneous progressive Markov process that has Borel measurable transition probabilities. In the case of a path-dependent diffusion process, the solutions to these integral equations lead to the concept of mild solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). Our goal is to establish uniqueness, stability, existence, and non-extendibility of solutions among a certain class of maps. By requiring the Feller property of the Markov process, we give weak conditions und…
