Search

We study the existence of nontrivial nonlocal nonnegative solutions $u(x,t)$ of the nonlinear initial value problems \[ (\partial_t -\Delta)^\alpha u\geq u^\lambda \quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0) \] and \[ C_1 u^\lambda \leq(\partial_t -\Delta)^\alpha u\leq C_2 u^\lambda \quad\text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0), \] where $\lambda,\alpha,C_1$, and $C_2$ are positive constants with $C_1

We investigate pointwise upper bounds for nonnegative solutions $u(x,t)$ of the nonlinear initial value problem \begin{equation}\label{0.1} 0\leq(\partial_t-\Delta)^\alpha u\leq u^\lambda \quad\text{ in }\mathbb{R}^n \times\mathbb{R},\,n\geq1, \end{equation} \begin{equation}\label{0.2} u=0\quad\text{in }\mathbb{R}^n\times(-\infty,0) \end{equation} where $\lambda$ and $\alpha$ are positive constants. To do this we first give a definition---tailored for our study of this problem---of fractional powers of the heat operator $(\partial_t-\Delta)^\alpha :Y\to X$ where $X$ and $Y$ are linear spaces whose elements are real valued functions on $\mathbb{R}^n \times\mathbb{R}$ and $0<\alpha<\alpha_0$ for some $\alpha_0$ which depends on $n$, $X$ and $Y$. We then obtain, when they exist, optimal pointwise upper bounds on $\mathbb{R}^n \times(0,\infty)$ for nonnegative solutions $u\in Y$ of this initial value problem with particular emphasis on those bounds as $t\to0^+$ and as $t\to\infty$., Comment: 38 pages, one figure

We study the behavior as $t\to 0^+$ of nonnegative functions \begin{equation}\label{0.1} u\in C^{2,1} (\mathbb{R}^n\times (0,1)) \cap L^\lambda (\mathbb{R}^n\times (0,1)),\quad n\ge 1, \end{equation} satisfying the parabolic Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq u_t-\Delta u\leq(\Phi^{\alpha/n}*u^\lambda )u^\sigma \quad \text{ in }B_1 (0)\times (0,1) \end{equation} where $\alpha\in(0,n+2)$, $\lambda>0$, and $\sigma\geq0$ are constants, $\Phi$ is the heat kernel, and $*$ is the convolution operation in $\mathbb{R}^n\times (0,1)$. We provide optimal conditions on $\alpha,\lambda$, and $\sigma$ such that nonnegative solutions $u$ satisfy pointwise bounds in compact subsets of $B_1(0)$ as $t\to 0^+$. We obtain similar results for nonnegative solutions when $\Phi^{\alpha/n}$ is replaced with the fundamental solution $\Phi_\alpha$ of the fractional heat operator $(\frac{\partial}{\partial t}-\Delta)^{\alpha/2}$., Comment: 40 pages

We study the behavior near the origin in $\mathbb{R}^n ,n\geq3$, of nonnegative functions \begin{equation}\label{0.1} u\in C^2 (\mathbb{R}^n \backslash \{0\})\cap L^\lambda (\mathbb{R}^n ) \end{equation} satisfying the Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq-\Delta u\leq(|x|^{-\alpha}*u^\lambda )u^\sigma \quad\text{ in }B_2 (0)\backslash \{0\} \end{equation} where $\alpha\in(0,n),\lambda>0,$ and $\sigma\geq0$ are constants and $*$ is the convolution operation in $\mathbb{R}^n$. We provide optimal conditions on $\alpha,\lambda$, and $\sigma$ such that nonnegative solutions $u$ satisfy pointwise bounds near the origin., Comment: 24 pages, 3 figures

We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. Question. For which continuous functions $f\colon [0,\infty)\to [0,\infty)$ does there exist a continuous function $\varphi\colon (0,1)\to (0,\infty)$ such that every $C^{2m}$ nonnegative solution $u(x)$ of $ 0 \le -\Delta^m u\le f(u)\quad$ in $B_2(0)\setminus\{0\}\subset {\bf R}^n $ satisfies $u(x) = O(\varphi(|x|))\quad \text{as } x\to 0 $ and what is the optimal such $\varphi$ when one exists?, Comment: 32 pages. This is a much-improved version of "Isolated Singularities of Nonlinear Polyharmonic Inequalities" arXiv:1107.4586

We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-\Delta u\leq v^\lambda\] \[0\leq v_t-\Delta v\leq u^\sigma\] in $\Omega\times (0,1)$, where $\lambda$ and $\sigma$ are nonnegative constants and $\Omega$ is an open subset of ${\mathbb R}^n$, $n\ge 1$. We provide optimal conditions on $\lambda$ and $\sigma$ such that solutions of this system satisfy pointwise bounds in compact subsets of $\Omega$ as $t\to 0^+$. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials., Comment: 31 pages, one figure

We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\le -\Delta u \le (\frac{1}{|x|^\alpha}* v)^\lambda$ $0\le -\Delta v \le (\frac{1}{|x|^\beta}* u)^\sigma$ in $B_2(0)\setminus\{0\} \subset R^n$, $n\ge 3$, where $\lambda,\sigma \ge 0$ and $\alpha,\beta\in (0,n)$. A by-product of our methods used to study these solutions will be results on the behavior near the origin of $L^1(B_1(0))$ solutions $f$ and $g$ of the system $0 \le f(x) \le C(|x|^{2-\alpha} + \int_{|y|<1}\frac{ g(y) dy}{|x-y|^{\alpha-2}} )^\lambda$ $0 \le g(x) \le C(|x|^{2-\beta} + \int_{|y|<1}\frac{ f(y) dy}{|x-y|^{\beta-2}} )^\sigma$ for $0<|x|<1$ where $\lambda,\sigma \ge 0$ and $\alpha, \beta\in (2,n+2)$., Comment: 32 pages, one figure. To be published in Calculus of Variations and PDE. Errors are corrected in the second paragraph of Step 2 on pages 22 and 23. Also corrections are made on page 27, line 9 from the bottom, and on page 29, line 3 from the bottom

We study classical positive solutions of the biharmonic inequality $-\Delta^2 v \geq f(v)$ in exterior domains in $\mathbb{R}^n$ where $f:(0,\infty)\to (0,\infty)$ is continuous function. We give lower bounds on the growth of $f(s)$ at $s=0$ and/or $s=\infty$ such that this inequality has no $C^4$ positive solution in any exterior domain of $\mathbb R^n$. Similar results were obtained by Armstrong and Sirakov [ Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (2011) 2011-2047] for $-\Delta v\ge f(v)$ using a method which depends only on properties related to the maximum principle. Since the maximum principle does not hold for the biharmonic operator, we adopt a different approach which relies on a new representation formula and an a priori pointwise bound for nonnegative solutions of $-\Delta^2u \ge 0$ in a punctured neighborhood of the origin in $\mathbb{R}^n$., Comment: 36 pages

We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\leq -\Delta u\leq f(v)$ $0\leq -\Delta v\leq g(u)$ in $B_1(0)\backslash\{0\}$ where $f,g:(0,\infty)\to (0,\infty)$ are continuous functions. We provide optimal conditions on $f$ and $g$ at $\infty$ such that solutions of this system satisfy pointwise bounds near the origin. In dimension $n=2$ we show that this property holds if $\log^+ f$ or $\log^+g$ grow at most linearly at infinity. In dimension $n\geq 3$ and under the assumption $f(t)=O(t^\lambda)$, $g(t)=O(t^\sigma)$ as $t\to \infty$, ($\lambda, \sigma\geq 0$), we obtain a new critical curve that optimally describes the existence of such pointwise bounds. Our approach relies in part on sharp estimates of nonlinear potentials which appear naturally in this context., Comment: 41 pages, 1 figure

We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. {\bf Question.} For which continuous functions $f\colon [0,\infty)\to [0,\infty)$ does there exist a continuous function $\phi\colon (0,1)\to (0,\infty)$ such that every $C^{2m}$ nonnegative solution $u(x)$ of 0 \le -\Delta^m u\le f(u)\quad \text{in}\quad B_2(0)\backslash\{0\}\subset {\bb R}^n satisfies u(x) = O(\phi(|x|))\quad \text{as}\quad x\to 0 and what is the optimal such $\phi$ when one exists?, Comment: 31 pages

We study the semilinear elliptic inequality $-\Delta u\geq\varphi(\delta_K(x))f(u)$ in $R^N\setminus K,$ where $\varphi, f$ are non-negative and continuous functions, $K\subset R^N$ $(N\geq 2)$ is a compact set and $\delta_K(x)={\rm dist}(x,\partial K)$. We obtain optimal conditions in terms of $\varphi$ and $f$ for the existence of $C^2$ positive solutions. Our analysis emphasizes the role played by the geometry of the compact set $K$.

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-\Delta^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these solutions $u$ satisfy a pointwise a priori bound as $x\to 0$., Comment: 18 pages