Local existence for evolution equations with nonlocal term in time and singular initial data

Aparcana A.; Castillo R.; Guzmán-Rea O.; Loayza M. · Zeitschrift fur Angewandte Mathematik und Physik (2022) · DOI: 10.1007/s00033-022-01723-x

Resumen

We consider the semilinear equation ut+(-Δ)α/2u=∫0tm(t,s)f(u(s))dsin Ω × (0 , T) , where 0 < α≤ 2 , m is a nonnegative and measurable homogeneous function defined on K= { (t, s) ∈ R2, 0 < s< t} , f is a nonnegative, continuous and nondecreasing function and Ω is either a bounded smooth domain or the whole space RN. Our goal is to determine conditions for the local existence and nonexistence of solutions with nonnegative initial data belonging to the space Lr(Ω) , 1 ≤ r< ∞.

Instituciones: Universidad Nacional San Luis Gonzaga