$A\Phi (t)+\int\limits {0} {t}{K(x,t-\tau )\Phi }(\tau )d\tau =H$

$A\Phi (t)+\int\limits_{0}^{t}{K(x,t-\tau )\Phi }(\tau )d\tau =H$
${{a}_{2}},{{a}_{3}}\ne 0$.

$K(t,\tau )$

${{\varphi }_{0j}}(t)=0$
${{\psi }_{nj}}(t)=0$
${{B}_{ij}}\times [0,T]$
${{\varphi }_{0j}}(t),{{\psi }_{nj}}(t)$

${{u}_{i0}}(x)\in C({{B}_{i}})$,${{f}_{i}}(x,t)\in ({{C}^{0,1}}\times \left[ 0,T \right])$ for $i=\overline{1,3}$,${{\varphi }_{i}}(t)$ and ${{\psi }_{i}}(t)$ are differentiable functions on $\left[ 0,T \right]$.Then problem (1), (4) has unique solution on $\left[ 0,T \right]$.

, .




, .

$({{L}_{2}},{{L}_{3}}>0)$.
$i=\overline{1,3}$ $\frac{1}{b_{1}^{2}}+\frac{1}{b_{2}^{2}}\ge 1$

,
Initial conditions
${{u}_{i}}(x,0)={{u}_{0i}}(x)$ $0<{{\alpha }_{i}}<1$
vertex conditions
${{u}_{1}}(0,t)={{a}_{1}}{{u}_{2}}(0,t)={{a}_{2}}{{u}_{3}}(0,t)$
${{u}_{1x}}(0,t)={{u}_{2x}}(0,t)+{{u}_{3x}}(0,t)$
${{u}_{1xx}}(0,t)-\frac{1}{{{a}_{1}}}{{u}_{2xx}}(0,t)-\frac{1}{{{a}_{2}}}{{u}_{3xx}}(0,t)=0$
${{u}_{i}}({{L}_{i}},t)={{\varphi }_{i}}(t)$ ${{u}_{jx}}({{L}_{j}},t)={{\psi }_{j}}(t)$
$i=\overline{1,3}$ $j=\overline{2,3}$
${{u}_{i}}\partial _{0t}^{{{\alpha }_{i}}}{{u}_{i}}\ge \frac{1}{2}\partial _{0t}^{{{\alpha }_{i}}}u_{i}^{2}$

$\frac{1}{2}\int\limits_{{{B}_{i}}}{\partial _{0t}^{{{\alpha }_{i}}}u_{i}^{2}}dx\le \int\limits_{{{B}_{i}}}{{{u}_{i}}{{u}_{ixxx}}dx+\int\limits_{{{B}_{i}}}{{{u}_{i}}{{f}_{i}}dx}}$

$\frac{1}{2}\sum\limits_{i=1}^{3}{\partial _{0t}^{{{\alpha }_{i}}}\int\limits_{{{B}_{i}}}{u_{i}^{2}dx\le \sum\limits_{i=1}^{3}{{{f}_{i}}{{u}_{i}}dx}}}$

$\sum\limits_{i=1}^{3}{(\frac{1}{2}-\varepsilon \Gamma (1-{{\alpha }_{i}}))D_{0t}^{{{\alpha }_{i}}-1}\int\limits_{{{B}_{i}}}{u_{i}^{2}(x,t)dx\le \frac{1}{4\varepsilon }\sum\limits_{i=1}^{3}{D_{0t}^{{{\alpha }_{i}}+1}\int\limits_{{{B}_{i}}}{f_{i}^{2}(x,t)}+\sum\limits_{i=1}^{3}{\frac{{{t}^{1-{{\alpha }_{i}}}}{{u}_{i}}(x,0)}{\Gamma (2-{{\alpha }_{i}})}}}}}$

$\varepsilon $

$G_{\alpha }^{\frac{2\alpha }{3}}(x,t)=\frac{1}{3{{t}^{1-\frac{2\alpha }{3}}}}\left\{ \begin{align}

& \phi (-\frac{\alpha }{3},\frac{2\alpha }{3};\frac{x}{{{t}^{\frac{\alpha }{3}}}}),x<0 \\
& -2\operatorname{Re}\left[ {{e}^{\frac{2\pi i}{3}}}\phi (-\frac{\alpha }{3},\frac{2\alpha }{3};{{e}^{\frac{2\pi i}{3}}}\frac{x}{{{t}^{\frac{\alpha }{3}}}}) \right],x>0 \\
\end{align} \right.$
$\frac{\partial }{\partial x}V_{\alpha }^{\frac{2\alpha }{3}}(x,t)=V_{\alpha }^{\frac{\alpha }{3}}(x,t)=\frac{1}{3{{t}^{1-\frac{\alpha }{3}}}}\operatorname{Im}\left[ {{e}^{\frac{4\pi i}{3}}}\phi (-\frac{\alpha }{3},\frac{\alpha }{3};{{e}^{\frac{2\pi i}{3}}}\frac{x}{{{t}^{\frac{\alpha }{3}}}}) \right],x>0$
$\partial _{0t}^{\alpha }G_{\sigma }^{\mu }(x,t)=G_{\sigma }^{\mu -\alpha }(x,t)$, $\frac{{{\partial }^{3}}}{\partial {{x}^{3}}}G_{\sigma }^{\mu }(x,t)=G_{\sigma }^{\mu -\sigma }(x,t)$
$\left| \partial _{0t}^{\alpha }G_{\sigma }^{\mu }(x,t) \right|\le C{{x}^{-\theta }}{{t}^{\mu +\theta \frac{\sigma }{3}-1}}$
$\theta \ge \left\{ \begin{align}
& 0,\text{ }(-\mu )\notin {{N}_{0}} \\
& 1,\text{ }(-\mu )\in {{N}_{0}} \\
\end{align} \right.$
${{w}_{1}}(x,t)=\int\limits_{0}^{t}{G_{\alpha }^{\frac{2\alpha }{3}}}(x-a,t-\eta ){{\tau }_{1}}(\eta )d\eta $
${{w}_{2}}(x,t)=\int\limits_{0}^{t}{V_{\alpha }^{\frac{2\alpha }{3}}}(x-a,t-\eta ){{\tau }_{2}}(\eta )d\eta $
${{w}_{3}}(x,t)=\int\limits_{0}^{t}{\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}G_{\alpha }^{\frac{2\alpha }{3}}}(x-a,t-\eta ){{\tau }_{3}}(\eta )d\eta $
${{w}_{4}}(x,t)=\int\limits_{0}^{t}{\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}V_{\alpha }^{\frac{2\alpha }{3}}}(x-a,t-\eta ){{\tau }_{4}}(\eta )d\eta $
${{w}_{5}}(x,t)=\int\limits_{0}^{t}{G_{\alpha }^{\frac{2\alpha }{3}}}(x-\xi ,t){{\tau }_{5}}(\xi )d\xi $
${{w}_{6}}(x,t)=\int\limits_{0}^{t}{\int\limits_{a}^{b}{G_{\alpha }^{\frac{2\alpha }{3}}}(x-\xi ,t-\eta )f(\xi ,\eta )d\xi d\eta }$

${{\tau }_{k}}(t)$ $k=1,2$ $(0,+\infty )$

.Functions ${{w}_{1}}(x,t)$ and ${{w}_{2}}(x,t)$ are solutions of the equation
$\partial _{0t}^{\alpha }{{u}_{i}}(x,t)-\frac{{{\partial }^{3}}{{u}_{i}}(x,t)}{\partial {{x}^{3}}}=0$;
2. Functions ${{w}_{1}}(x,t)$ and ${{w}_{2}}(x,t)$ satisfy conditions
$\underset{t\to 0}{\mathop{\lim }}\,{{w}_{k}}(x,t)=0,k=1,2.$