Well-posedness and stability for a class of solutions of semi-linear diffusion equations with rough coefficients
Pham Truong Xuan, Le The Sac
TL;DR
We address well-posedness and stability of pseudo almost periodic mild solutions for semi-linear diffusion equations with rough coefficients by developing an abstract parabolic framework on interpolation spaces. The linear problem is shown to admit a PAP-preserving solution operator under polynomial decay bounds $\|e^{-tA}Bv\|$, enabling a fixed-point argument to obtain a unique PAP mild solution to $u'(t)+Au(t)=BG(u)(t)$. The theory is then applied to heat-type equations with rough coefficients, notably $u'(t)-b\Delta u(t)=|u|^{m-1}u+F(t)$, establishing existence, uniqueness, and polynomial stability for $F\in PAP$, with explicit decay in $L^{r,\infty}$ spaces; this extends prior work by allowing $0\in\sigma(A)$ and rough coefficients. Overall, the results provide a robust PAP-based framework for semi-linear diffusion problems beyond exponential stability, with implications for diffusion and fluid-dynamic models.
Abstract
In this work we study the existence, uniqueness and polynomial stability of the pseudo almost periodic mild solutions of semi-linear diffusion equations with rough coefficients in certain interpolation spaces. First, we rewirte the equations in abstract parabolic equation. Then, we use the polynomial stability of the semigroups of the corresponding linear equations to prove the boundedness of the solution operator for the linear equations in appropriate interpolation spaces. We show that this operator preserves the pseudo almost periodic property of functions. We will use the fixed point argument to obtain the existence and stability of the pseudo almost periodic mild solutions for the semi-linear equations. The abstract results will be applied to the semi-linear diffusion equations with rough coefficients to obtain our desired results.