Abstract
A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.
Highlights
In the disk, D = {(x, y) ∈ R2, x2 + y2 < b2}, we consider the following evolution fractional partial differential equation in the Riemann–Liouville sense:∂tλU – U + δUt + A(x, y, t)U = h(x, y, t), ∀(x, y) ∈ D, t > 0, (1.1)with a damping effect due to the term δUt, with δ > 0, and where ∂tλU is the Riemann– Liouville fractional derivative of order λ ∈
We introduce the fractional derivative spaces Htη,x([0, T], ) and HTη,x([0, T], ) to be the space of functions U ∈ L2x([0, T], ) having η-order Riemann–Liouville derivative ∂tηU ∈ L2x([0, T], ), U(x, 0) = 0, U(x, T) = 0, having, respectively, the norms
The fractional derivative is considered in the Riemann–Liouville sense
Summary
D = {(x, y) ∈ R2, x2 + y2 < b2}, we consider the following evolution fractional partial differential equation in the Riemann–Liouville sense:. ∂tλU – U + δUt + A(x, y, t)U = h(x, y, t), ∀(x, y) ∈ D, t > 0, (1.1). With a damping effect due to the term δUt, with δ > 0, and where ∂tλU is the Riemann– Liouville fractional derivative of order λ ∈ The functions h(x, y, t) and A(x, y, t) are given and will be specified later. If we search only for the radially symmetric solution (the functions h and A must be radial), we have the partial differential equation ∂tλU 1 r (rUr )r + δUt. A(r, θ , t)U (r, t) =
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