{'fraction_non_alphanumeric': 0.06010452961672474, 'fraction_numerical': 0.010452961672473868, 'mean_word_length': 3.9313304721030042, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Mathematical Model for the Use of Energy Resources: a singular parabolic equation

We consider a singular parabolic equation tut − ∆u = f , for (x, t) ∈ Ω × (0, T ), arising in symmetric boundary layer flows. Here Ω ⊂ R is a bounded domain with C boundary ∂Ω, β ≤ 1, f = f(t, x) is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equation with Dirichlet boundary conditions, and we investigate its regularity. There are two different cases depending on β. If β < 1, for any initial data u0 ∈ L(Ω), there exists a unique weak solution, which in fact is a strong solution. The singularity is removable when β < 1. While if β = 1, there exists a unique solution of the singular parabolic problem tut −∆u = f . The initial data cannot be arbitrarily chosen. In fact, if f is continuous and f(t) → f0, as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem −∆u = f0, for x ∈ Ω, with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong.