论文标题
非线性时空分数扩散方程的解决方案的定性特性
Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation
论文作者
论文摘要
在本文中，我们研究了具有多项式非线性的非局部非线性扩散方程的凯奇 - 迪里奇问题$$ \ MATHCAL {d} _ {0 | T}^{α} u+（ - δ）^s_pu =γ| U |^{m-1} u+++μ| U | u | u | u | u |^{q-2} u，\ \，\，γ，γ，μ\ in \ in \ mathbb {r} $ \ MATHCAL {D} _ {0 | T}^α$和Space-Fractional $ p $ -laplacian Operator $（ - δ）^S_P $。我们简单地证明了使用纯粹的代数关系对考虑的问题的比较原则，对于$γ，μ，m $和$ q $。 Galerkin近似方法用于证明存在局部弱解的存在。使用比较原理对全球解决方案的爆炸现象，全球弱解决方案的存在和渐近行为进行了分类。
In the present paper, we study the Cauchy-Dirichlet problem to the nonlocal nonlinear diffusion equation with polynomial nonlinearities $$\mathcal{D}_{0|t}^{α}u+(-Δ)^s_pu=γ|u|^{m-1}u+μ|u|^{q-2}u,\,γ,μ\in\mathbb{R},\,m>0,q>1,$$ involving time-fractional Caputo derivative $\mathcal{D}_{0|t}^α$ and space-fractional $p$-Laplacian operator $(-Δ)^s_p$. We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of $γ,μ,m$ and $q$. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.
