The heat equation with singular nonlinearity and singular initial data

Abstract

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation u t − u = a(x)u q + b(x)u p in a bounded domain and with Dirichlet's condition on the boundary. We consider here a ∈ L α (Ω), b ∈ L β (Ω) and 0 < q 1 < p. The initial data u(0) = u 0 is considered in the space L r (Ω), r 1. In the main result (0 < q < 1), we assume a, b 0 a.e. in Ω and we assume that u 0 γ d Ω for some γ > 0. We find a unique solution in the space C([0, T ], L r (Ω)) ∩ L ∞ loc ((0, T ), L ∞ (Ω)).

Attractors for non-autonomous parabolic problems with singular initial data

“…When f (x, u) = λu + a(x)u p , p > 1, u 0 ∈ L ∞ (Ω) and u 0 0, the estimates of positive solutions of (1.6) with a(x) ∈ C (Ω) and the blow-up of solutions of (1.6) with a(x) ∈ C 2 (Ω) have been studied respectively in [36]. Recently, if f (x, u) = a(x)u q + b(x)u p , 0 < q 1 < p, a(x) ∈ L α (Ω), b(x) ∈ L β (Ω), α, β 1, it has been proved in [28] that there exists a unique positive solution u ∈ C [0, T ]; L r (Ω) ∩ L ∞ loc (0, T ); L ∞ (Ω) (1.7) of (1.6) with u 0 ∈ L r (Ω), 1 r < ∞, and u 0 γ d Ω , where γ is a positive constant, d Ω = dist(x, ∂Ω).…”

“…To obtain the existence of uniform attractors in L r (Ω) for the family of processes corresponding to (1.1), higher regularity of solutions of (1.1) than the results in [13,28] is needed. Since the nonlinearity f (x, u) and external force g(x, t) of (1.1) depend on x, and g(x, t) belongs to L p loc (R; X), which is equipped with the local p-power mean convergence topology different from the topology of X α associated to linear operator , − f (x, u) + g(x, t) is not an -regular map, and the abstract results for (1.10) in [6] cannot be applied directly to system (1.1).…”

Nonlocal Dynamic Problems with Singular Nonlinearities and Applications to MEMS

“…However, we will see that our approach allows more general nonlocal effects than the one considered here and in previous works [5,6,16,20,13,14]. We mention that evolution MEMS equations have been previously handled by different methods in [18,11,12], where existence results are obtained avoiding nonlocal contributions. More recently, results for nonlocal parabolic problems are obtained in [13] whereas the second-order hyperbolic nonlocal MEMS equation is studied in the one-dimensional case in [14].…”

Stability of stationary solutions for semilinear heat equations with concave nonlinearity

“…In [17], the nonuniqueness is pointed out, and moreover, the comparison principle is established only for positive solutions. The long-time behavior of solutions is also investigated by using the principle in [8,33], where semilinear heat equations with convexconcave nonlinearity are treated (see also related works [2,12,18,24,3,25,13,14]). The nonuniqueness of solution for (1.1)-(1.3) and major difficulties in this issue are arising from the concave nonlinearity, in particular, the fact that the nonlinear term is more singular at u = 0 than other semilinear heat equations that have been vigorously studied so far (e.g., reaction-diffusion equation and blow-up case).…”