On a problem for the nonlinear diffusion equation with conformable time derivative

Abstract

In this paper, we study a nonlinear diffusion equation with conformable derivative: ${\mathfrak{D}}_t^{\left(\alpha \right)}u-\mathrm{\Delta }u=\mathcal{L}(x,t;u(x,t\left)\right)$, where $0<\alpha <1,(x,t)\in \mathrm{\Omega }\times (0,T)$. We consider both of the problems:

• Initial value problem: the solution contains the integral $I={\int }_0^t{\tau }^{\gamma }\mathrm{d}\tau$ (critical as $\gamma \le -1$).

• Final value problem: not well-posed (if the solution exists it does not depend continuously on the given data).

For the initial value problem, the lack of convergence of the integral I, for $\gamma \le -1$. The existence for the solution is represented. For the final value problem, the Hadamard instability occurs, we propose two regularization methods to solve the nonlinear problem in case the source term is a Lipschitz function. The results of existence, uniqueness and stability of the regularized problem are obtained. We also develop some new techniques on functional analysis to propose regularity estimates of regularized solution.

Keywords:

• Conformable derivative
• existence
• regularity
• direct problems
• inverse problems

2010 Mathematics Subject Classifications:

• 26A33
• 34B16
• 47A52

Disclosure statement

No potential conflict of interest was reported by the author(s).