Well‐Posedness for Time‐Fractional Diffusion With Singular Multiplicative Forcing
Ngo Ngoc Hung
Source abstract
We consider a time‐fractional diffusion equation on ℝ with a possibly singular multiplicative forcing term. By exploiting an operator convolution identity, the original mild formulation is reduced to a Volterra integral equation involving the fractional diffusion family. Using standard Mittag–Leffler kernel estimates, scaling‐consistent L p ( ℝ )– L q ( ℝ ) smoothing bounds, and time‐weighted convolution estimates, we establish existence and uniqueness of mild solutions in suitable time‐weighted spaces. The forcing profile is allowed to be nonsmooth in space and to exhibit an integrable singularity near t = 0, whereas the nonlinearity is assumed to be globally Lipschitz with σ (0) = 0. We further prove that the solution attains the prescribed initial data in L q ( ℝ ) and depends continuously on both the initial data and the multiplicative forcing profile.
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