Existence of radially symmetric patterns for a diffusion problem with variable diffusivity

We give a sufficient condition for the existence of radially symmetric stable stationary solution of the problem ut = div(a2∇u)+ f (u) on the unit ball whose border is supplied with zero Neumann boundary condition. Such a condition involves the diffusivity function a and the technique used here is inspired by the work of E. Yanagida.


Introduction
In this paper we consider the radially symmetric stationary solutions of the problem where B = x ∈ R 2 : x < 1 , ν is the unit outer normal vector to ∂B, a(x) is a positive radially symmetric function in B (i.e.a(x) = a( x )) and f ∈ C 1 (R).
This kind of problem appears as a mathematical model in many distinct areas, for example: biological population growth process, selection-migration model or, more generally, any problem of the concentration of a diffusing substance in a heterogeneous medium whose diffusivity is a 2 (x), under the effect of the source or sink term f (u).
Stable non-constant stationary solutions to (1.1) are sometimes simply referred to as patterns.On the existence and non-existence of patterns for scalar diffusion equation there is a vast literature that we can summarize as follows: [3,9,10,15] in intervals; [5] in balls of R n ; [1,7,13,14] in surfaces of revolution or Riemannian manifold with or without boundary and [2,6,11] in bounded domains of R n .In particular, [2,11] consider constant diffusivity (i.e.a(x) =constant) and prove that there is no pattern if the domain is convex (B ⊂ R n , for instance) regardless of the function f .See also the references in these works.
Email: mcn.sonego@unifei.edu.br The technique used herein requires that the term of diffusivity a(x) be analytic in B, see Theorem 1.1 below.This hypothesis allows us to conclude two properties of a: a( x ) = a(r) is analytic in [0, 1] (recall that a is radially symmetric) and a (0) = 0 (throughout the text we use to denote the derivative in relation to r).Both play a key role in this work.
In order to present our main result, note that (1.1) is equivalent to the following one: since any radially symmetric solution of (1.1) must satisfy (1.2).Throughout the text, in many instances, we use (1.2) instead of (1.1).
Our main result can be stated as follows.
Theorem 1.1.If for some r 0 ∈ (0, 1) it holds that and if a(x) is analytic in B then there exists f ∈ C 1 (R) such that problem (1.1) admits a radially symmetric pattern.
In [5] do Nascimento considered the same problem and proved that if a 2 (r) satisfies r 2 a + ra − a ≤ 0 on (0, 1) then every non-constant stationary solution of (1.1) is unstable, i.e, there are no patterns.This result extends those obtained by Yanagida [15] and Hale et al. [3] in a interval (namely, a ≤ 0 and (a 2 ) ≤ 0 respectively).The Theorem 1.1 shows that if a is analytic in B (i.e.a(r) analytic in [0, 1]) then the condition obtained by do Nascimento is also necessary for non-existence of patterns (see Remark 3.1 (1)).To see this, simply expand (1.3), namely (ar) Undoubtedly, this was the main motivation of the present study.
Our proof follows the steps proposed in [15] where the problem is considered in an interval and it is proved that if a (r 0 ) > 0 for some r 0 in this interval then there exists f such that the corresponding problem possesses patterns.The same method has been adapted for problems on surfaces of revolution, see [1,8,13].In particular, Punzo [13] considered the problem on surfaces of revolution without boundary and some of his ideas were adapted here due to the close relationship of symmetry present in both problems.
The paper is divided as follows: in the Preliminaries we proof three essential lemmas for our method while Section 3 is dedicated to the proof of Theorem 1.1.

Preliminaries
We recall that by a stationary solution of problem (1.1) we mean a solution to the problem and for the linearized problem ((2.1) in a neighborhood of U) the sign of the principal eigenvalue λ 1 indicates the stability of U, i.e., if λ 1 > 0 then U is asymptotically stable and if λ 1 < 0 then U is unstable.If λ 1 = 0 then stability or instability can occur.This is so called linear stability and, roughly speaking, means that solutions of the corresponding parabolic equation (1.1) with the initial data near U will tend to U, as t → ∞.
Lemma 2.1.Let v be a radial solution of problem (1.1).Let there exist w Then v is asymptotically stable.
Proof.Let λ 1 be the principal eigenvalue of the linearized problem and let φ 1 be the corresponding eigenfunction.Then It follows that λ 1 > 0 and v is asymptotically stable.
Using (1.3) and the regularity of a, we can take 0 It is not difficult to see that also occurs Now, consider the linear ordinary differential equation where with K > 0 a parameter to be chosen later.Since a(r) is analytic in [0, 1] we can infer that the functions p(r) and q(r) are analytic in [0, 1] and p 0 := lim r→0 p(r) = 1 and q 0 := lim r→0 q(r) = −1.
The above steps -inspired by [13] where a problem on surfaces of revolution without boundary was considered -are to ensure that (2.8) Also consider z 2 = z 2 (r) a solution of the initial value problem z(1) = 0, z (1) = −1. (2.9) We can find K > 0 such that for every K ≥ K. Indeed, it suffices to note that and a (0) = 0. We shall write z i (r) = z i (r, K) (i = 1, 2) to indicate the dependence of the solution on the parameter K.
(4) Fix any K 1 > K.By integrating the equation (2.8), and remembering that z 1 is a solution, we get for any Integrating again where c 1 and c 2 are constants independent of K.As a > 0 and using the item (3) of this lemma, we obtain The claim follows by letting K → ∞.The proofs for (5)-( 8) are analogous.
For our next lemma we define the function z where z 3 is a positive smooth function such that z is smooth at the points r = R 2 and r = R 3 .Thus, z is smooth in [0, 1], z > 0 in (0, 1) and z(0) = z(1) = 0. Proof.The function u = Z(r) is increasing in (0, 1), since z > 0 in (0, 1).Hence we can define the inverse function (2.13) The rest of the proof follows by the same arguments as in the proof of Lemma 3.5 in [8].

Proof of the main theorem
This section is devoted to prove the Theorem 1.1.Let z be the function defined by (2.11) and m 1 , m 2 > 0 constants to be chosen later.Define We have that w = w(r, K, m 1 , m 2 ) (see (2.8), (2.9) and ( 2 ) and is positive in [0, 1].In order to use Lemma 2.1, we prove that if Z is a stationary solution of (1.1) defined by (2.12), then there exist m 1 > 0, m 2 > 0 and K > 0 such that L(w) ≡ (a 2 rw ) r + f (v)w ≤ 0 for r ∈ (0, 1) and w (1) > 0.
Note that this proves the Theorem 1.1 with f given by (2.13).
A simple but laborious calculation shows that (a 2 r(az) ) r = a (a 2 z) + a 2 z r − az (ar) r and then In order to conclude that the term between brackets above is zero, simply derive the equation and recall that Z = z.