201319

Code: 201319

Regularity of Solutions to Nonlinear Time Fractional Differential Equation

Mirjana Stojanovic

 

Abstracts

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space <img height="31" border="0" style="vertical-align:bottom" width="408" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0252960213601186-si1.gif">$C^{\infty }\left(t\in \left(0,\infty \right);H_2^{s+2}\left({\text{R}}^n\right)\right)\cap C^0\left(t\in [0,\infty \right);H^s\left(R^n\right)),\hspace{0.17em}s\hspace{0.17em}\in R,$ to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions

equation(0.1)

<img height="54" border="0" style="vertical-align:bottom" width="304" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0252960213601186-si2.gif">$\begin{array}{l}{\int }_0^{2}p\left(\beta \right)D_{*}^{\beta }\hspace{0.17em}u\left(x,t\right)d\beta ={\Delta }_{x}u\left(x,t\right)+f(t,u(t,\hspace{0.17em}x)),\\ t\ge 0,x\in {\text{R}}^n,u\left(o,x\right)=u_t(0,x)=\psi (x),\end{array}$

where Δ_x is the spatial Laplace operator, <img height="22" border="0" style="vertical-align:bottom" width="24" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0252960213601186-si3.gif">$D_{*}^{\beta }$ is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval <img height="27" border="0" style="vertical-align:bottom" width="502" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0252960213601186-si4.gif">$(0,2)\hspace{0.17em}i.e.,\hspace{0.17em}p(\beta )=\sum _{k=1}^m{b}_k\hspace{0.17em}\delta \left(\beta -{\beta }_k\right),\hspace{0.17em}0\hspace{0.17em}<{\beta }_k<2,\hspace{0.17em}{b}_k>0,\hspace{0.17em}k=1,2,\mathrm{\dots },m.$ The regularity of the solution is established in the framework of the space C^∞(t∈(0,∞); C^∞(Rn))∩Co(t∈[0,∞);C^∞(Rn))$C^{\infty }\left(t\in \left(0,\infty \right);\hspace{0.17em}{C}^{\infty }\left(R^n\right)\right)\cap C^o(t\in [0,\infty );C^{\infty }(R^n))$ when the initial data belong to the Sobolev space