Comportement asymptotique des solutions globales pour quelques problèmes paraboliques non linéaires singuliers

Abstract : In this thesis, we study the nonlinear parabolic equation ∂ t u = ∆u + a |x|⎺⥾ |u|ᵅ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, ⍺ ∈ R, α > 0, 0 < Ƴ < min(2,N) and with initial value u(0) = φ. We establish local well-posedness in Lq(Rᴺ) and in Cₒ(Rᴺ). In particular, the value q = N ⍺/(2 − γ) plays a critical role.For ⍺ > (2 − γ)/N, we show the existence of global self-similar solutions with initial values φ(x) = ω(x) |x|−(2−γ)/⍺, where ω ∈ L∞(Rᴺ) is homogeneous of degree 0 and ||ω||∞ is sufficiently small. We then prove that if φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺ for |x| large, then the solution is global and is asymptotic in the L∞-norm to a self-similar solution of the nonlinear equation. While if φ(x)∼ω(x) |x| (x)|x|−σ for |x| large with (2 − γ)/α < σ < N, then the solution is global but is asymptotic in the L∞-norm toe t(ω(x) |x|−σ). The equation with more general potential, ∂ t u = ∆u + V(x) |u|ᵅ u, V(x) |x |⥾ ∈ L∞(Rᴺ), is also studied. In particular, for initial data φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺, |x| large , we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity. we study also the nonlinear parabolic system ∂ t u = ∆u + a |x|⎺⥾ |v|ᴾ⎺¹v, ∂ t v = ∆v + b |x|⎺ ᴾ |u|q⎺¹ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, a,b ∈ R, 0 < y < min(2,N)? 0 < p < min(2,N), p,q > 1. Under conditions on the parameters p, q, γ and ρ we show the existence and uniqueness of global solutions for initial values small with respect of some norms. In particular, we show the existence of self-similar solutions with initial value Φ = (φ₁, φ₂), where φ₁, φ₂ are homogeneous initial data. We also prove that some global solutions are asymptotic for large time to self-similar solutions. As a second objective we consider the nonlinear heat equation ut = ∆u + |u|ᴾ⎺¹u - |u| q⎺¹u, where t ≥ 0 and x ∈ Ω, the unit ball of Rᴺ, N ≥ 3, with Dirichlet boundary conditions. Let h be a radially symmetric, sign-changing stationary solution of (E). We prove that the solution of (E) with initial value λ h blows up in finite time if |λ − 1| > 0 is sufficiently small and if 1 < q < p < Ps = N+2/N−2 and p sufficiently close to Ps. This proves that the set of initial data for which the solution is global is not star-shaped around 0.
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Byrame Ben Slimene. Comportement asymptotique des solutions globales pour quelques problèmes paraboliques non linéaires singuliers. Mathématiques générales [math.GM]. Université Sorbonne Paris Cité; Université de Tunis El Manar, 2017. Français. ⟨NNT : 2017USPCD059⟩. ⟨tel-02290647⟩

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