论文标题
具有非线性卷积术语的高阶进化不平等
Higher order evolution inequalities with nonlinear convolution terms
论文作者
论文摘要
我们关注的是对$$解决方案的存在和不存在的研究{case}＆\ displayStyle \ frac {\ partial^k u} {\ partial t^k}+（ - eartial t^k}+（ - Δ） \ Mathbb r _+，\\ [0.1in]＆\ displayStyle \ frac {\ partial^i u} {\ partial t^i}（x，0）= u_i（x，x）\，\，\，\，\，\，\ text {in} $ n，k，m \ geq 1 $是正整数，$ p，q> 0 $和$ u_i \ in l^1 _ {\ rm loc}（\ Mathbb {r}^n）$ for $ 0 \ leq i \ leq i \ leq k-1 $。我们假设$ k $是一个径向的正且连续的功能，它在无穷大街区降低。在上面的问题中，$ k \ ast | u |^p $表示$ k（| x |）$和$ | u |^p $之间的标准卷积操作。我们在$ N，M，K，P $和$ Q $上获得必要的条件，以便上述问题具有解决方案。我们的分析强调了$ \ displaystyle \ frac {\ partial^{k-1} u} {\ partial t^{k-1}} $的作用。
We are concerned with the study of existence and nonexistence of weak solutions to $$ \begin{cases} &\displaystyle \frac{\partial^k u}{\partial t^k}+(-Δ)^m u\geq (K\ast |u|^p)|u|^q \quad\mbox{ in } \mathbb R^N \times \mathbb R_+,\\[0.1in] &\displaystyle \frac{\partial^i u}{\partial t^i}(x,0) = u_i(x) \,\, \text{ in } \mathbb R^N,\, 0\leq i\leq k-1,\\ \end{cases} $$ where $N,k,m\geq 1$ are positive integers, $p,q>0$ and $u_i\in L^1_{\rm loc}(\mathbb{R}^N)$ for $0\leq i\leq k-1$. We assume that $K$ is a radial positive and continuous function which decreases in a neighbourhood of infinity. In the above problem, $K\ast |u|^p$ denotes the standard convolution operation between $K(|x|)$ and $|u|^p$. We obtain necessary conditions on $N,m,k,p$ and $q$ such that the above problem has solutions. Our analysis emphasizes the role played by the sign of $\displaystyle \frac{\partial^{k-1} u}{\partial t^{k-1}}$.
