论文标题
具有结构性阻尼和非本地性非线性的时空分数半线性方程的爆破溶液
Blowing-up solutions of a time-space fractional semi-linear equation with a structural damping and a nonlocal in time nonlinearity
论文作者
论文摘要
在本文中，我们使用时间空间分数结构阻尼和时间非局限性的时间空间\ begin \ begin {equation*} {\ mathbf {d}} _ {0 | t}^{1 +α_1} u + +α_1} u + + +α_1} u + + +α_1} u + +α_1} （-Δ）^σu+（ - δ）^δ\ Mathbf {d} _ {0 | t}^{α_2} u = i_ = i_ {0 | t}^{1-γ} | u | u |^{p} \ end {equation*}其中$ p> 1 $，$α_i，γin（0,1）$，$δ，σ\ in（0,1）$，$ {\ Mathbf {d}} _ {0 | t}^t}^{t}^{α_I} $是前2次批准的和$ iis $ i______________} 0 | 0 | Riemann-Liouville订单$1-γ$的分数积分。如果\开始{等式*} 1 <p \ leqslant \ frac {2（2+α_1-γ）} {（\ frac {\ frac {α_1+1}σn+2γ-2α_1-2）_+1，$ niste $ n preace*我们将结果扩展到系统\ begin {align*}＆{\ MathBf {d}} _ {0 | t}^{1 +α_1} u +（-Δ） i_ {0 | t}^{1-γ_{1}} | v |^{p}，\ qquad（t，x）\ in（0，\ infty）\ times \ times \ times \ mathbb {r}^n，\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ＆{\ MathBf {d}} _ {0 | t}^{1+β_{1}} v+（ - δ） I_{0|t}^{1-γ_2}|u|^{q},\qquad (t,x)\in (0,\infty )\times \mathbb{R}^{N}, \end{align*} where $p,q>1$, $0<δ_i$, $σ_i<1$ and $γ_2\in （0,1）$。此外，我们提出了存在本地或全球解决方案的必要条件。
In this paper, we investigate the semilinear equation with a time-space fractional structural damping and a nonlocal in time nonlinearity \begin{equation*} {\mathbf{D}}_{0|t}^{1+α_1}u + (-Δ)^σu+(-Δ)^δ\mathbf{D}_{0|t}^{α_2} u = I_{0|t}^{1-γ}|u|^{p}, \qquad (t,x)\in (0,\infty) \times \mathbb{R}^N, \end{equation*} where $p>1$, $α_i, γ in (0,1)$, $δ, σ\in (0,1)$, ${\mathbf{D}}_{0|t}^{α_i}$ is the Caputo fractional derivative and $I_{0|t}^{1-γ}$ is the Riemann-Liouville fractional integral of order $1-γ$. We prove the non-existence of global solutions if \begin{equation*} 1<p\leqslant \frac{2(2+α_1-γ)}{(\frac{α_1+1}σ N+2γ-2α_1-2)_+ }+1, \end{equation*} for any space dimension $N\geqslant 1.$ Then, we extend the result to the system \begin{align*} &{\mathbf{D}}_{0|t}^{1+α_1}u + (-Δ)^{σ_1} u + (-Δ)^{δ_1}{\mathbf{D}}_{0|t}^{α_2} u = I_{0|t}^{1-γ_{1}}|v|^{p},\qquad (t,x)\in (0,\infty) \times \mathbb{R}^N, \\ &{\mathbf{D}}_{0|t}^{1+β_{1}}v+(-Δ)^{σ_2} v + (-Δ)^{δ_2}{\mathbf{D}}_{0|t}^{β_2}v = I_{0|t}^{1-γ_2}|u|^{q},\qquad (t,x)\in (0,\infty )\times \mathbb{R}^{N}, \end{align*} where $p,q>1$, $0<δ_i$, $σ_i<1$ and $γ_2\in (0,1)$. Also, we present the necessary conditions for the existence of local or global solutions.
