Stability/nonstability properties of renormalized/entropy solutions for degenerate parabolic equations with $$L^{1}$$
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Stability/nonstability properties of renormalized/entropy solutions for degenerate parabolic equations with L1 /measure data M. Abdellaoui1 Received: 7 April 2020 / Accepted: 23 May 2020 © Sociedad Española de Matemática Aplicada 2020
Abstract We study the possibility to give a formulation to the degenerate parabolic problems modeled by u t − div |∇u| p−2 ∇u)/(1 + |u|)θ ( p−1) = μ in (0, T ) × , 1 (Pb ) u(0, x) = u 0 (x) in , u(t, x) = 0 on (0, T ) × ∂, where θ > 0, u 0 ∈ L 1 () and μ is a general (nonnegative) Radon measure. We also investigate the strong stability of solutions for noncoercive absorption problems whose model u t − div |∇u| p−2 ∇u)/(1 + |u|)θ ( p−1) + |u|q−1 u = f in (0, T ) × , 2 (Pb ) u(0, x) = 0 in , u(t, x) = 0 on (0, T ) × ∂ where q > r ( p − 1)[1 + θ ( p − 1)]/(r − p) and f ∈ L 1loc (Q\K ) with K is a compact subset of Q of zero r -capacity (or, is a measure concentrated on a set of r -capacity zero). We prove the convergence of approximate solutions u n (related to a regular approximation μn of μ) towards a renormalized solutions u of (Pb1 ), and we extend the previous known-results on the nonstability of entropy solutions for problems (Pb2 ). Keywords Concentration phenomena · Generalized solutions · Degenerate parabolic equations · Approximate methods · Measures · Capacities Mathematics Subject Classification 35A01, 35D30, 35K65, 35A20, 28A12
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M. Abdellaoui [email protected] LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, B.P. 1796, Atlas Fez, Morocco
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M. Abdellaoui
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Some basic facts on capacity and measures . . . . . . 2.1 Notations and definitions . . . . . . . . . . . . . 2.2 Capacity and measures . . . . . . . . . . . . . . 3 Assumptions and statement of the first stability result . 4 Proof of the stability result . . . . . . . . . . . . . . . 5 Strong stability result for parabolic problem (1.2) . . . 6 Main results and Proofs . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction In this paper we study the stability and non-stability results for degenerate parabolic problems in a domain Q = (0, T ) × ( is a bounded open subset of R N , N ≥ 2, and T > 0) with right-hand side in L 1loc (Q) (the space of locally integrable functions) or M(Q) (the space of, general, Radon measures with bounded total variation). Surprisingly, although (in general) the finite energy (weak) solu
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