带反应项的一维退化扩散方程解的正则性
Regularity of Solutions to One-Dimensional Degenerate Diffusion Equations with Reactions
AI总结：
该研究针对含反应项的一维退化扩散方程，建立了解与自由边界的系统正则性理论，强化了自由边界运动速度的已有结论，并将多孔介质方程的经典正则性性质推广至更广的方程类。
AI中文摘要：
我们研究一维反应扩散方程\( u_t=[A(u)]_{xx}+f(x,u) \)。该扩散算子属于一类广泛的非线性退化扩散算子，其中包含多孔介质算子作为特例。我们针对解及其自由边界建立了系统的正则性理论：首先，建立压力变量\( v \)的\( C^1 \)正则性，并给出其二阶空间导数的下界；其次，证明达西定律，且在等待时间后，右（对应左）自由边界以严格正（对应负）速度运动（定理3.1），从而强化了此前仅给出非负（对应非正）性的已知结果；最后，在扩散项与反应项的附加结构假设下，得到解及其自由边界的更高阶正则性（定理4.5和4.6）。这些结果将多孔介质方程的若干经典正则性性质推广到更广泛的带反应项的退化扩散方程类中。
英文摘要：
We study the one-dimensional reaction-diffusion equation \[ u_t=[A(u)]_{xx}+f(x,u). \] The diffusion operator belongs to a broad class of nonlinear degenerate diffusion operators that includes the porous medium operator as a special case. We develop a systematic regularity theory for the solutions and their free boundaries. First, we establish the $C^1$ regularity of the pressure variable $v$, together with a lower bound for its second spatial derivative. Next, we prove Darcy's law and that, after the waiting time, a right (resp.\ left) free boundary moves with strictly positive (resp.\ negative) velocity (Theorem 3.1), thereby strengthening the previously known results which only gave nonnegativity (resp.\ nonpositivity). Finally, under additional structural assumptions on the diffusion and reaction terms, we obtain higher regularity for both the solution and its free boundaries (Theorems 4.5 and 4.6). These results extend several classical regularity properties of the porous medium equation to a much broader class of degenerate diffusion equations with reactions.
