论文标题
非线性分数抛物线不平等的非负溶液的存在
Existence of Nonnegative Solutions of Nonlinear Fractional Parabolic Inequalities
论文作者
论文摘要
我们研究了非线性非本地非负溶液的存在$ u（x，t）$的非线性初始值问题\ [（（\ partial_t-Δ）^αu\ geq u \ geq u^geq u^λ\ quad \ quad \ quad \ quad \ quad {in} \ quad \ text {in} \ mathbb {r}^n \ times（ - \ infty，0）\] \ times \ mathbb {r}，\，n \ geq1 \] \ [u = 0 \ quad \ text {in} \ mathbb {r}^n \ times（ - \ infty，0），\]，其中$λ，α，α，α，c_1 $和$ c_2 $和$ c_2 $与$ c_1 $ c_1 <c_1 <c_1我们使用[taliaferro，2020]中给出的分数热运算符$（\ partial_t-Δ）^α$的定义，并将我们在经典案例$α= 1 $中比较我们的结果与已知结果。
We study the existence of nontrivial nonlocal nonnegative solutions $u(x,t)$ of the nonlinear initial value problems \[ (\partial_t -Δ)^αu\geq u^λ\quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0) \] and \[ C_1 u^λ\leq(\partial_t -Δ)^αu\leq C_2 u^λ\quad\text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0), \] where $λ,α,C_1$, and $C_2$ are positive constants with $C_1 <C_2$. We use the definition of the fractional heat operator $(\partial_t -Δ)^α$ given in [Taliaferro, 2020] and compare our results in the classical case $α=1$ to known results.
