论文标题
准自相似性及其应用于超线性热方程的全局可溶性
Quasi self-similarity and its application to the global in time solvability of a superlinear heat equation
论文作者
论文摘要
本文涉及半线性热方程\ begin {equation}的全球时间存在解决方案 \ tag {p} \ label {eq：p} \ begin {case} \ partial_t u =ΔU + f（u）， ＆x \ in \ mathbb {r}^n，\，\，\，t> 0， \\ [3pt] u（x，0）= u_0（x）\ ge 0， ＆x \ in \ mathbb {r}^n， \ end {cases} \ end {equation}其中$ n \ ge 1 $，$ u_0 $是非负初始函数，$ f \ in C^1（[0，\ infty））\ cap c^2（（（0，\ infty））$表示问题的超级非线性。我们考虑问题的全局存在和解决方案的不存在，问题〜\ eqref {eq：p}。本文的主要目的是确定解决方案全球存在的初始功能的关键衰减率。特别是，我们表明它的特征是准自相似解决方案，该解决方案是解决方案$ w $ \ of begin {equation} \ notag ΔW + \ frac {y} {2} \ cdot \ nabla w + f（w）f（w） + f（w） + \ frac {| \ nabla w |^2} {f（w）f（w）} \ bigl [ Q -f'（w）f（w） \ bigr] = 0， \ Quad y \ in \ mathbb {r}^n，\ end {qore}其中$ f（s）：= \ displayStyle \ int_s^{\ infty} \ dfrac {1} {f（f（η）}dη$和$ q \ ge 1 $。
This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = Δu + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0, &x\in \mathbb{R}^N, \end{cases} \end{equation} where $N\ge 1$, $u_0$ is a nonnegative initial function and $f\in C^1([0,\infty)) \cap C^2((0,\infty))$ denotes superlinear nonlinearity of the problem. We consider the global in time existence and nonexistence of solutions for problem~\eqref{eq:P}. The main purpose of this paper is to determine the critical decay rate of initial functions for the global existence of solutions. In particular, we show that it is characterized by quasi self-similar solutions which are solutions $W$ of \begin{equation} \notag ΔW + \frac{y}{2}\cdot \nabla W + f(W)F(W) + f(W) + \frac{|\nabla W|^2}{f(W)F(W)} \Bigl[ q - f'(W)F(W) \Bigr] = 0, \quad y \in \mathbb{R}^N, \end{equation} where $F(s):=\displaystyle\int_s^{\infty}\dfrac{1}{f(η)}dη$ and $q\ge 1$.
