Linear Non-Autonomous Heat Flow in $$L_0^1({ {\mathbb {R}}}^{d})$$ and Applications to Elliptic Equations in $${ {\mathbb {R}}}^{d}$$
Linear Non-Autonomous Heat Flow in $$L_0^1({ {\mathbb {R}}}^{d})$$ and Applications to Elliptic Equations in $${ {\mathbb {R}}}^{d}$$
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$$L_0^1({ {mathbb {R}}}^{d})$$中的线性非自治热流及其在$${ {mathbb {R}}}^{d}中椭圆方程的应用
DOI：
10.1007/s10884-022-10195-6
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发表时间：
2022
影响因子：
1.3
通讯作者：
Robinson J
中科院分区：
文献类型：
--
作者：
Robinson J
我们研究了方程的解，对于初始数据是“在无穷远处大”，就像我们以前关于非强迫热方程的论文中所处理的那样。当我们将那些解收敛于0的特征化为，并不是所有的初始数据都能达到这一点。当我们给出条件以保证解由通常的“常数变分公式”u(t)=e-λ ts (t)u0+∫0t -λ(t- S)S(t- S)f(S)ds给出时，其中为热半群。我们用这些结果来处理椭圆型问题，当f被允许为“无穷大”时，给出了一个解存在的条件，该解是通过与通常的格林函数卷积得到的。我们的许多结果都是尖锐的。
We study solutions of the equation, for initial data that is ‘large at infinity’ as treated in our previous papers on the unforced heat equation. Whenwe characterise thosefor which solutions converge to 0 as, as not everyis able to achieve that for all initial data. Whenwe give conditions to guarantee that the solution is given by the usual ‘variation of constants formula’ u(t)=e-λtS(t)u0+∫0te-λ(t-s)S(t-s)f(s)ds, whereis the heat semigroup. We use these results to treat the elliptic problemwhenfis allowed to be ‘large at infinity’, giving conditions under which a solution exists that is given by convolution with the usual Green’s function for the problem. Many of our results are sharp when.