Initial-boundary value problems to semilinear multi-term fractional differential equations

Abstract

For $ν,ν_i,μ_j\in(0,1)$, we analyze the semilinear integro-differential equation on the one-dimensional domain $Ω=(a,b)$ in the unknown $u=u(x,t)$ \[ \mathbf{D}_{t}^ν(\varrho_{0}u)+\sum_{i=1}^{M}\mathbf{D}_{t}^{ν_{i}}(\varrho_{i}u) -\sum_{j=1}^{N}\mathbf{D}_{t}^{μ_{j}}(γ_{j}u) -\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u)=g(x,t), \] where $\mathbf{D}_{t}^ν,\mathbf{D}_{t}^{ν_{i}}, \mathbf{D}_{t}^{μ_{j}}$ are Caputo fractional derivatives, $\varrho_0=\varrho_0(t)>0,$ $\varrho_{i}=\varrho_{i}(t)$, $γ_{j}=γ_{j}(t)$, $\mathcal{L}_{k}$ are uniform elliptic operators with time-dependent smooth coefficients, $\mathcal{K}$ is a summable convolution kernel. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under certain structural conditions on the nonlinearity $f$ and orders $ν,ν_i,μ_j$, the global existence and uniqueness of classical and strong solutions to the related initial-boundary value problems are established via the so-called continuation arguments method. The crucial point is searching suitable a priori estimates of the solution in the fractional Hölder and Sobolev spaces. The problems are also studied from the numerical point of view.

Figures (3)

• Figure 1: The behavior of the function $\omega_{1-\nu}(t)$ and the kernels $\mathcal{N}(t;\nu,\frac{\nu}{j+1})$ for $j=1,2,3,$ with (a) $T^{*}=0.01$, (b) $T^{*}=0.1$.
• Figure 2: Solutions to Example \ref{['e.2']} with $f(x,t,u) = 0$, $\nu_1=\nu/3$, $\mu_1=\nu/2$.
• Figure 3: Solutions to Example \ref{['e.2']} with $f(x,t,u) = x t \cos(u^2)$, $\nu_1=\nu/3$, $\mu_1=\nu/2$.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Theorem 4.1
• Remark 4.2
• Definition 4.3
• Theorem 4.4
• Remark 4.5