By Alfonso Castro

The authors offer an entire type of the radial suggestions to a category of response diffusion equations coming up within the research of thermal constructions equivalent to plasmas with thermal equilibrium or no flux on the boundary. particularly, their examine comprises quickly turning out to be nonlinearities, that's, these the place an exponent exceeds the severe exponent. They describe the corresponding bifurcation diagrams and make sure lifestyles and area of expertise of floor states, which play a vital position in characterizing these diagrams. in addition they supply info at the stability-unstability of the radial regular states

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**Additional info for Classification of Radial Solutions Arising in the Study of Thermal Structures with Thermal Equilibrium or No Flux at the Boundary**

**Example text**

Let us see now that ξ can be extended to (1, D). 1) 1 ¯ d¯ + δ) with v(1, ξ(d), d) = 1}. = sup{δ > 0; ξ can be extended to (d, If d¯+ 1 < D, by Lemma 8 and Theorem 11 we have that v(·, 1, d¯+ 1 ) − 1 oscillates. Let τ be the k-th zero of v(·, 1, d¯+ 1 ) − 1. Arguing as in the previous paragraph we see that there exists η > 0 and a continuous function ξ1 : (d¯+ 1 − η, d¯ + 1 + η) → (0, ∞) such that v(ξ1 (d), 1, d) − 1 = 0 and ξ1 (d¯+ 1 ) = τ . By the non-degeneracy of the zeroes, ξ1 (d) is the k-th zero of v(·, 1, d) − 1.

It follows from Theorem 9 that σ(1, d1 ) = ∞. 12) on initial data we obtain that limd→D ξj (d) = ∞, j = 2, 3, . . Case −1 < q < 1. We split the proof in two subcases: −1 < q ≤ 0 and 0 < q < 1. ´ A. CASTRO AND V. PADRON 50 Subcase −1 < q ≤ 0. There exist positive constants m1 , m2 such that m1 ≤ f (v)/(v − 1) ≤ m2 for any v ∈ (0, D]. Thus by the Sturm comparison theorem there exists of cj such that 0 < tj (d) ≤ cj for d ∈ (1, D), and j = 1, 2, . . Subcase 0 < q < 1. 5 and Theorem 1 we know that tj (d) is uniformly bounded on any compact subset of [1, D).

When k → ∞ we obtain that r 1 ¯ ¯ ) − f (τ ) dτ, k = 0, 1, . . 6). Therefore, by uniqueness H ¯ ≡ K. 11) ¯ ≤ K(r), r ∈ [0, w(0)]. 11). Hence there exists t ∈ (0, M ) such that v(t) = w(t). Suppose now that v(a) = w(a) = ρ > 1 for some a > 0, and v(t) > w(t) for 0 ≤ t < a. By uniqueness of solutions to initial values problems v (a) < w (a). Since ρ > 1 and (f (v)/(v − 1)) ≤ 0 on (1, ∞) (see lemma 22 below), it follows that (w(t) − 1)f (v(t)) − (v(t) − 1)f (w(t)) ≤ 0, 0 ≤ t ≤ a. 1), this inequality can be written as [t(−(w(t) − 1)v (t) + (v(t) − 1)w (t))] ≤ 0, 0 ≤ t ≤ a.