A nonlinear parabolic problem with singular terms and nonregular data

Abstract

A. We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the formwhere Ω is an open bounded subset of R N (N ≥ 2), 0 is a nonnegative integrable function, ∆ is the -laplace operator, µ is a nonnegative bounded Radon measure on Ω × (0 T ) and is a nonnegative function of L 1 (Ω × (0 T )). The term is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing .

A parabolic problem involving $p(x)$-Laplacian, a power and a singular nonlinearity

“…Bonis & Giachetti in [10], assumed the functions f and g to be in some L r (0, T ; L m (Ω)) space with 1 r + N pm < 1. In the same spirit of [10], in [32] the authors have shown the existence of a weak solution by considering g to be in L 1 (Q T ) and by replacing the function f with a bounded Radon measure µ. The parabolic problems involving p(x)-Laplacian and a measure data or an L 1 data (the case λ = 0 and g ≡ 0 of (1.1)) have been analyzed by several authors since the papers [5,38,42].…”

“…Let us now discuss some of the important parabolic problems which has helped us in the construction of this work. Concerning the case λ = 0 and p(•) = p of (1.1), the existence result has been investigated in [10,11,12,32] and the bibliography therein. Bonis & Giachetti in [10], assumed the functions f and g to be in some L r (0, T ; L m (Ω)) space with 1 r + N pm < 1.…”

On nonlinear parabolic equations with singular lower order term

“…In the same concept the authors in [23] proved the existence of solution to problem with 𝛾 > 0, f is a nonnegative function on , and is a nonnegative bounded Radon measures on . Hence Charkaoui and Alaa [7] established the existence of weak periodic solution to singular parabolic problems with 𝛾 > 0 and f is a nonnegative integrable function periodic in time with period T. Let us observe that we refer to [8,9,11,17,24] for more details on singular parabolic problems.…”

On some parabolic equations involving superlinear singular gradient terms

“…Here, the authors prove existence of a distributional solution via an approximation argument and one of the main tools is a suitable 2 application of the Harnack inequality in order to deduce the positivity of the approximating sequence. More recently, in presence of a general h and measure data, existence and uniqueness has been addressed in [31], under suitable assumptions. When g(0) = +∞, h ≡ 1 and for some 1 < q < p, problem (1.1) has been already analyzed in [14] for bounded initial data u 0 and for f satisfying the Aronson-Serrin curve.…”