Abstract
We examine finite-time blow-up solutions toin a ball , , where D and S generalize the functionswith . We show that if as well as and is a nonnegative, radially symmetric classical solution to () blowing up at , then there exists a so-called blow-up profile satisfyingMoreover, for all withwe can find such thatfor all .
Highlights
The possibility of blow-up constitutes one of the most striking features of the quasilinear system uvtt = = ∇· ∆v (D(u, −v+v)∇u u, S(u, v)∇v),(uD(·(, u0), v)∇u − S(u, = u0, v(·, 0) v)∇v) = v0, · ν = ∂ν v0, in Ω × (0, T ), in Ω × (0, T ), on ∂Ω × (0, T ), in Ω
The reasoning from [39], where estimates on blow-up profiles to solutions to (KS) with D ≡ 1 and S(u, v) = u have been derived, is to consider w := ζαu with ζ(x) ≈ |x| and to make use of semi-group arguments as well as Lp-Lq estimates in order to derive an L∞ bound for w which in turn implies the desired estimate of the form (1.12) for u
The existence of blow-up profiles is shown in Section 4 by considering global solutions, ε ∈ (0, 1), to suitably approximative problems which converge on all compact sets in Ω \ {0} × (0, ∞) to (u, v) for certain functions u, v : Ω × [0, ∞) → [0, ∞). We prove that these functions coincide which u and v on Ω × [0, Tmax) such that we may set U := u(·, Tmax) as well as V := v(·, Tmax) and make use of regularity of u and v
Summary
The possibility of (finite-time) blow-up constitutes one of the most striking features of the quasilinear system. The reasoning from [39], where estimates on blow-up profiles to solutions to (KS) with D ≡ 1 and S(u, v) = u have been derived, is to consider w := ζαu with ζ(x) ≈ |x| and to make use of semi-group arguments as well as Lp-Lq estimates in order to derive an L∞ bound for w which in turn implies the desired estimate of the form (1.12) for u Through their mere nature, these methods are inadequate to handle equations with nonlinear diffusion. The existence of blow-up profiles is shown in Section 4 by considering global solutions (uε, vε), ε ∈ (0, 1), to suitably approximative problems which converge (along a subsequence) on all compact sets in Ω \ {0} × (0, ∞) to (u, v) for certain functions u, v : Ω × [0, ∞) → [0, ∞).
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