STABILITY RESULTS FOR DISCONTINUOUS NONLINEAR ELLIPTIC AND PARABOLIC PROBLEMS WITH A S-SHAPED BIFURCATION BRANCH OF STATIONARY SOLUTIONS

We study stability of the nonnegative solutions of a discontinuous elliptic eigenvalue problem relevant in several applications as for instance in climate modeling. After giving the explicit expresion of the S-shaped bifurcation diagram (λ, ‖uλ‖∞) we show the instability of the decreasing part of the bifurcation curve and the stability of the increasing part. This extends to the case of non-smooth nonlinear terms the well known 1971 result by M.G. Crandall and P.H. Rabinowitz concerning differentiable nonlinear terms. We point out that, in general, there is a lacking of uniquenees of solutions for the associated parabolic problem. Nevertheless, for nondegenerate solutions (crossing the discontinuity value of u in a transversal way) the comparison principle and the uniqueness of solutions hold. The instability is obtained trough a linearization process leading to an eigenvalue problem in which a Dirac delta distribution appears as a coefficient of the differential operator. The stability proof uses a suitable change of variables, the continuuity of the bifurcation branch and the comparison principle for nondegenerate solutions of the parabolic problem.

1. Introduction.We consider in this paper the nonlinear eigenvalue problem associated to nonnegative solutions of the discontinuous elliptic equation in (0, 1), where λ > 0 and f (u) is given by
Our interest is on nonnegative solutions.In fact, as we shall see later, nonnegative solutions must be strictly concave functions and thus such that max x∈[0,1] |u(x)| = u(0) and u > 0 on [0, 1).A solution u λ of problem P (λ, f ) is a function u ∈ C 2 ((0, 1)−{x µ,λ })∩C 1 ([0, 1)), for some x µ,λ ∈ [0, 1) where u(x µ,λ ) = µ (called as the free boundary associated to u) and with u ≥ 0, u = 0, such that −u (x) = λf (u(x)), for any x ∈ (0, 1) − {x µ,λ }, and u (0) = 0, u(1) = 0. Problem P (λ, f ) can be considered as a simplified version of some more general formulations arising in several different contexts.We emphasize that the assumption (1.2) is crucial since the nature of models and solutions for the case of f 0 > 1 is entirely different (see, e.g.[10], [39] and [6]).Problems dealing with the case f 0 ∈ (0, 1) arise, for instance, in the study of chemical reactors and porous media combustion (see e.g., [22], [25], [26], [24]), steady vortex rings in an ideal fluid ( [27]), plasma studies ( [41], [30], [19]), the primitive equations of the atmosphere in presence of vapor saturation ( [8]), etc.Besides the above applications, our special main motivation was the consideration of problem P (λ, f ) as a simplified version of the so called diffusive energy balance models arising in climatology (see, e.g.[37], [34], [11], [39], [21] and a stochastic version in [17]).Although these models must be formulated on a Riemannian manifold without boundary representing the Earth atmosphere [21], the so called 1d-model corresponds to the case in which the surface temperature is assumed to depend only on the latitude component.By neglecting the term modeling the emitted terrestrial energy flux we lead to a formulation similar to P (λ, f ) in which the spatial domain (0, 1) must be associated to a semisphere, the discontinuous function represents the co-albedo (with a discontinuity which is associated to the radical change of the co-albedo when the temperature is crossing −10 centigrade degrees), the parameter λ the so-called solar constant, the boundary condition u (0) = 0 formulates the simplified assumption of symmetry between both semispheres and the condition u(1) = 0 represents the renormalized temperature at the North pole (i.e.we are assuming that u = T + T N where T N < −10 represents the North pole temperature and thus µ = −10 − T N > 0).Results on the asymptotic behaviour, when t → +∞, for the evolution energy balance model were obtained in [15] (see also [43]) where it was also proved the general multiplicity of stationary solutions according the value of λ.A sharper bifurcation diagram, as a S-shaped curve was rigorously obtained in [1].Nevertheless the method of proof in [1] uses the information obtained trough suitable zero-dimensional energy balance models and thus there is lacking of a more detailed information about the associated free boundaries generated by the solutions (given as the spatial points where T = T N ).A next paper by the authors shall be devoted to the extension of the results of this paper to the case in which the absorption term, modeling the emitted terrestrial energy flux, is taken into account.This additional appears also in the simplification made by McKean [36] of the initial value problem for the FitzHugh-Nagumo equations which were introduced as a model for the conduction of electrical impulses in the nerve axon (see, e.g., Termam [42]).
In Section 2 of this paper we shall obtain an explicit S-shaped bifurcation curve for the nontrivial nonnegative solutions of P (λ, f ).Although there are several results in the previous literature that allow us to conclude easily that the bifurcation curve (λ, u λ ∞ ) must be S-shaped (see, e.g.[40]), in order to carry out our stability study we shall present the explicit expression of such a curve as well as the information about the free boundary associated to solutions u λ according the values of λ: i.e. the points x µ,λ ∈ [0, 1) where u λ (x µ,λ ) = µ.
By denoting u ∞ = max |u(x)|, we shall prove, the following result: f0 then there exists a unique solution u * λ without free boundary of P (λ, f ).Moreover i.e. the line (λ, γ * (λ)) defines an increasing part of the bifurcation diagram.
iii) If λ ∈ (λ 0 , λ 1 ] then there exists u λ solution of P (λ, f ) with a free boundary given by Moreover, Moreover its free boundary is given by and Note that the behaviour of the branch near the two "turning points" (λ 0 , u λ0 ) and (λ 1 , u * λ1 ) is different and does not coincide with the results on the subject for the case of f (u) smooth: in the first case lim λ λ0 dγ(λ) dλ = +∞ and lim λ λ0 dγ(λ) dλ = −∞ but in the second one 0 < lim λ λ1 dγ(λ) dλ < +∞ and lim λ λ1 dγ(λ) dλ < f0 2 .Qualitatively, the associated bifurcation branch is similar to the one represented in Figure 1 below.
The stability of solutions u λ and u λ will be analyzed in Sections 3 and 4 respectively.We recall that in the case of smooth nonlinear functions f (u) the instability of the decreasing part of the bifurcation curve (i.e. of solutions u λ ) and the stability of the increasing part (i.e. of solutions u * λ and u λ ) was shown in the famous paper Crandall and Rabinowitz [9] (for the application to the so called Sellers energy balance model with a smooth co-albedo function f (u) see Hetzer [34]).One of our main goals in this paper is to prove that the same type of conclusions remains true for the case of non-smooth functions f (u) by using some ad hoc methods.Before to present a brief idea of the rest of the Sections we point out that the solutions u λ of P (λ, f ) are also solutions of the multivalued problem where β is the maximal monotone graph of R 2 given by As explained later, both problems are equivalent in the class of nonnegative solutions thanks to the special case of those boundary conditions (as already mentioned, they lead to strictly concave functions).One reason to reformulate problem P (λ, f ) as P * (λ, f ) is because the associated parabolic problem x ∈ (0, 1), is well-posed (under suitable conditions on v 0 ) in contrast to what may happen with the associated parabolic version of the discontinuous problem (see, e.g. the comments presented in [45]).
In Section 3 we shall collect results on the solvability of problem P P * (λ, β, v 0 , ).We shall adapt to our framework some of the results of [11] and [21] showing that P P * (λ, β, v 0 , ) may have a multiplicity of solutions for some initial datum v 0 (x) and that, nevertheless, the solution is unique (and the comparison principle holds) in the class of "non degenerate solutions": i.e. solutions v such that for any θ ∈ (0, θ 0 ) and for any t > 0, for some C > 0 and θ 0 > 0.Here meas(.)denotes the Lebesgue measure.
In Section 4, we shall prove the instability of stationary solutions u λ of the decreasing part of the branch of solutions (λ, γ(λ)).If we denote simply by x µ,λ ∈ (0, 1) the free boundary generated by u λ (i.e.u λ (x µ,λ ) = µ) then we shall show that the instability of u λ is an easy consequence of the study of the eigenvalue problem associated to the linearized equation x ∈ (0, 1) Here δ {x µ,λ } denotes the Dirac delta distribution at the free boundary point x µ,λ .
We point out that eigenvalue problems with some measures as coefficients of the operators (similarly to P ν (x µ,λ : f 0 , λ)) arise in several completely different contexts (see, e.g. the survey by Belloni and Robinett [3] on "quantum dots" for linear Schrödinger equations and some stability studies for KKP equations presented in Liang, Li and Matano [35]).For some nonlinear eigenvalue problems with measures in the operators see [18] and [12].
The stability of the increasing parts of the bifurcation curve, (λ, γ(λ)) and (λ, γ * (λ)), will be proved in Section 5 by a different technique to the linearization argument.We shall not construct, neither, any Lyapunov function.Our method of proof will use the comparison principle for the parabolic problem (in the class of non-degenerate solutions) and a suitable change of variables involving the parameter λ.This adapts to our framework some ideas of the papers [13] and [6].We define with v(t, x) solution of P P * (λ, β, v 0 , ).Then V satisfies A similar change of variable leads to the new stationary problem In this way, by proving the continuous dependence of solutions of P (L, f ) with respect to L, we shall be able to show that given a λ ∈ (λ 0 , +∞) the solutions u λ for λ near λ lead to sub and supersolutions of the parabolic problem P P (v 0 , λ) which are closed enough to u λ .A similar argument (even easier) applies to the part (λ, γ * (λ)) of the branch.This implies the stability of the solutions in the increasing parts of the branch.
Theorem 1.3.Solutions u λ and u * λ are L ∞ −stable Finally, we point out that it seems possible to study the H 1 −stability of solutions of P P * (λ, β, v 0 ) for different values of λ, by using other type of techniques (see, e.g.Arrieta, Rodríguez-Bernal and Valero [2], for a related discontinuous problem with λ prescribed and different boundary conditions, and Díaz, Hernández and Ilyasov [14], for a different related nonlinear free boundary eigenvalue problem).This goal will be presented in a different paper by the authors.
The main result of this Section is Theorem 1.1 stated in the Introduction.
Proof of Theorem 1.1.i) First we consider the easier case of solutions such that f (u(x)) = f 0 for any x ∈ [0, 1] (i.e. with absence of free boundary).Then, since −u (x) = λf 0 an easy calculation shows that Hence, denoting this solution by u * , we obtain (1.3) since max In the rest of the proof we shall search solutions u with a free boundary x µ,λ ∈ [0, 1) and we consider the corresponding problems verified by u on the different regions (0, x µ,λ ) and (x µ,λ , 1).On (0, x µ,λ ) we get the linear problem and then On (x µ,λ , 1) we get and thus with B := f0 2−f0 .In order to study this condition (giving information on the multiplicity and location of the free boundary) let us introduce the function for r ∈ (0, 1).
3. On the multivalued parabolic problem.Concerning the parabolic problem P P * (λ, β, v 0 , ), it is not too difficult to adapt to this setting some previous results in the literature (see, e.g.[24], [11] and [21]) concerning similar diffusion operators and other boundary conditions.We introduce the energy space It is well known that V is a closed subspace of the Hilbert space H 1 (0, 1).Given v 0 ∈ L 2 (0, 1) (more general initial data can be also considered: see, e.g.[7]) with v 0 ≥ 0 a.e. on (0, 1), the notion of solution we shall use in this paper is the following: )) for a.e.(t, x) ∈ (0, ∞) × (0, 1), such that for every test function ζ ∈ L 2 (0, ∞ : V ) with ζ t ∈ L 2 (0, ∞ : V ) and ζ(t, .)= 0 with compact support on (0, ∞), we have that v(0, .)= v 0 (.) in L 2 (0, 1) and Concerning the existence of solutions we have: Proof.Parts i) and iii) are obvious adaptations of Theorem 1 and Corollary 1 of [11] (see also [21]).Part ii) was proved in [16] for the case of the diffusion operator associated to a Riemannian manifold without boundary (the particularization to our framework is a routine matter since the diffusion operator is now much more simple to be treated on the space X = C([0, 1]).
We point out that, since the initial datum v 0 does not need to be a strictly concave function, there is no equivalence, in general, among solutions of the multivalued and discontinuous parabolic equations associated trough the process of filling the jump.
be the unique solution of P (λ 1 , f ) without free boundary.Then problem P P * (λ 1 , β, v 0 ) admits at least two different solutions.
Remark 1.Some different nonuniqueness results could be found by using selfsimilar special solutions as in Gianni and Hulshof [29].
To avoid free boundaries such that x µ (t) ↑ +∞ as t ↓ 0 leading to non-uniqueness results we need to impose some kind of condition to the solutions v saying that when v is "crossing" the discontinuity value µ they do that in a "transversal way".This is the reason to introduce the following notion of "nondegeneracy property".Definition 3.3.We say that a function v(x) satisfies the nondegeneracy property at the level µ if there exists C > 0 and θ 0 > 0 such that for any θ ∈ (0, θ 0 ), meas{x ∈ (0, 1) such that |v(x) − µ| ≤ θ} ≤ Cθ, where meas(.)denotes the Lebesgue measure on (0, 1).
The following result gives the comparison of solutions (and thus the uniqueness of solutions) in the class of solutions satisfying the non-degeneracy property.Theorem 3.4.Let v 0 , v 0 , v 0 ∈ V such that v 0 (x) ≤ v 0 (x) ≤ v 0 (x) a.e.x ∈ (0, 1).

By Poincaré and Sobolev inequalities
where C 1 > 0. Then by Lemma 3.1 we get Then, if we conclude that which leads to the conclusion.If (3.14) does not hold we introduce the rescaling y = αx with α > 0. Given a general function h(x, t) we define h(y, t) by h(y, t) = h(αx, t).Then the functions v(y, t) and v(y, t) satisfy ∂v ∂t − α 2 v yy = λz(y, t), ∂v ∂t − α 2 v yy = λẑ(y, t), in (0, α) × (0, T ).Then, as in ( [11], [21]) it is easy to see that by taking α large enough we get to a new negative balance between the involved constants (as in (3.14)) and thus the conclusion holds.
Remark 3. Several additional results on non-degeneracy solutions for the case of other boundary conditions can be adapte to the present framewor.In particular, it can be shown that if v 0 ∈ V is non-degenerate then there exists a (unique) nondegenerate solution v(x, t) (see, [11], [21], [20] and their references).
4. Instability of the lower solution.
Definition 4.1.Given the solution u λ (x) of problem P (λ, f ), we say that u λ is L ∞ −stable if ∀ > 0, ∃δ > 0 such that for any v 0 ∈ L ∞ (0, 1) generating a nondegenerate solution v(t, .: v 0 ) of P P * (λ, β, v 0 ) and verifying that It is well-known that in the case of Lipschitz or singular nonlinear functions f (u) the stability and instability of a solution of the stationary problem u λ is reduced to study the sign of the principal eigenvalue of the linearized problem (see, e.g. Henry [32] and Hernández, Mancebo and Vega [33]).In our setting this would lead to consider the problem x ∈ (0, 1), = 0.
In the case of the discontinuous function f (u) given by (1.1) we observe that if x µ,λ ∈ (0, 1) denotes the free boundary generated by u λ (i.e.u λ (x µ,λ ) = µ) then and thus ) where δ {x µ,λ } denotes the Dirac delta distribution at the free boundary point x µ,λ .Then the linearized problem becomes problem P ν (x µ,λ : f 0 , λ) defined in the Introduction.
By a solution of the problem P ν (x µ,λ : f 0 , λ) we mean a function U ∈ V with the condition that U is discontinuous in x µ,λ and U ∈ V and satisfying the equation in V .We also recall that we say that an eigenvalue ν = ν 1 is the "principal eigenvalue" of P ν (x µ,λ : f 0 , λ) if that problem admits a strict positive solution on (0, 1).
We point out that the proof of the fact that if the principal eigenvalue of P ν (x µ,λ : f 0 , λ) is negative then the stationary solution is unstable (such as given, for instance, in [32]) remains valid even if f (u λ (x)) involves a measure as in (4.16).Essentially, the main idea of the proof is to consider solutions v of the parabolic problem P P * (λ, β, v 0 ) with v 0 = u λ +ηw 0 with w 0 smooth and η small enough and to approximate v, as η → 0, by functions of the form v(x, t) = u λ (x) + ηw(x, t) with w(t, x) = e −νt U (x), U solution of the eigenvalue problem P ν (x µ,λ : f 0 , λ).
In this section we shall prove the instability of stationary solutions u λ of the decreasing part of the branch of solutions (λ, γ(λ)) by proving that the principal eigenvalue of P ν (x µ,λ : f 0 , λ) is negative.Before to present the proof of Theorem 1.2 we point out that, such as it was indicated in Fleishman and Mahar [26] for the case of Dirichlet boundary conditions, it is possible to characterize the solutions of P ν (x µ,λ : f 0 , λ) in terms of a suitable transmission problem.Proposition 4.1.Let U be a solution of P ν (x µ,λ : ε, λ).Denote by U ± (x µ,λ ) and U ± (x µ,λ ) the directional limits of U and U on x = x µ,λ .Then U satisfies , Moreover, the converse implication is valid.
Proof.By well-known regularity results U ∈ C 2 ((0, 1) − {x µ,λ }) ∩ C 0 ([0, 1]) and thus satisfies −U (x) = νU (x) when x = x µ,λ .To prove the jump condition on U we consider ζ ∈ V be a test function function satisfying We consider a regularization of the distribution δ {x µ,λ } by a sequence of smooth functions h n (x) so that h n ∈ C 0 ([0, 1]) and h n → δ {x µ,λ } in the set of bounded Radon measures M b (0, 1).Let U n be the (unique) solution of the problem Since (P υ,n ) is a linear eigenvalue problem we can renormalize the eigenfunctions by assuming, for instance, U n ∞ = 1 for any n ∈ N.Moreover, which implies (since by well-known regularity results U n ∈ H 2 (0, 1)) (4.17) Observe that U n is uniformly bounded in C 0 ([0, 1]) and thus U n is uniformly bounded in BV (0, 1).So, there is subsequence, still denoted by U n , and a function U ∈ C 0 ([0, 1]) with U n ∈ BV (0, 1) such that U n U in M b (0, 1) and U n → U in BV (0, 1) and U n → U in C 0 ([0, 1]) as n → +∞.Integrating by parts in (4.17) Taking limits, first as l → 0, k → 0, and then as n → +∞ we obtain the stated jump conditions (since U n → U in BV (0, 1) implies the convergence of the directional derivatives in the point x µ,λ ).The proof of the reciprocal implication reduces merely to write the definition of U as an element of V .
Remark 4. The instability of the stationary solutions of the decreasing part of the branch of solutions when we replace our boundary conditions by the Dirichlet boundary conditions was announced, without any proof, in the paper Fleishman and Mahar [26].They claim that a proof of the negativeness of the principal eigenvalue of the associated problem could be obtained by using a characterization of the eigenvalue problem in terms of a transmission problem similar to (P ν ), as indicated in Proposition 4.1.Nevertheless, as far as we know such a proof was never published by them.In fact, our proof of Theorem 1.2 will not use this fact and so it is absolutely independent of their claim (no study of the associated parabolic problem is done in the mentioned reference).
Proof of Theorem 1.2.Since we search to prove that for ν < 0 there is a positive solution we write ν = −τ 2 .Since necessarily Then, there exists τ * > 0 such that Hence, from Lemma 4.3, we conclude that This leads to the conclusion if we have x µ,λ ∈ (0, 1  2 ).Now, it remains to prove that τ 1 > 0 also in the case x µ,λ ∈ ( 1 2 , 1 2−f0 ).In that case it is clear that there exists If we take now τ * f0 defined as a positive solution of the equation then, from Lemma 4.3, we conclude that τ 1 > τ * f0 > 0. Summarizing, since we have proved that the solution τ 1 of (4.19) is strictly positive, τ 1 > 0, to end the proof of Theorem 1.2 it is enough to take B > 0, arbitrary, and then A > 0 given by (4.18).In this way we are building a positive solution U ∈ C 0 ((0, 1)) and thus the principal eigenvalue of the problem (P ν ) satisfies ν 1 = −τ 2 1 < 0 and, hence, the lower branch γ(λ) is an unstable branch of solutions.and thus Since, we search a C 1 function with a single free boundary, then for x > x γ , u γ must satisfy In particular, u(x γ ) ≥ 0 if and only if x ∈ [0, L(γ)], with i.e., L(γ) is given by (5.20).On the other hand, it is clear that u γ (x) is not degenerate.Indeed, and since u γ ∈ C 1 and is strictly concave Thus, the set of points where |u γ (x) − µ| ≤ θ is contained in the interval of measure 2θ/ 2(γ − µ).
We recall that we have an explicit description of the function γ(λ).Then, by the change of variable we get L(λ) = λ(γ), where λ(γ) is the inverse function of γ(λ).
Then, since both V ( x, t) and u γ+h ( x) are non-degenerate and since both functions are continuous on (0, L(λ)), we conclude from Theorem 3.4 that V ( x, t) ≤ u γ+h ( x), t > 0.
Analogously, we have also the comparison from above, i.e. u γ−h ( x) ≤ V ( x, t).
Then, by Lemma 5.2, we get that V (., t) − u λ (.) L ∞ (0,L(λ)) ≤ , which ends the proof after making, again, the change of variables x = √ λx.The proof of the stability of the increasing part of the branch (λ, γ * (λ)) is similar, and even easier than before since in this part of the branch the solutions do not have free boundary (since u * L ∞ (0,1) < µ) and we can take a neighborhood of u * in L ∞ (0, 1) such that the solutions with initial datum in it do not have free boundary.In consequence they are solutions of a linear problem for which the L ∞ −stability is well-known.
Remark 5.The above arguments have some common points with the ones used in the paper by Bertsch and Klaver [6].Nevertheless, we point out that their stability results only apply (for the one-dimensional case) when f 0 > 1 (see their Theorem 7.1) and thus assumption (1.2) is not satisfied.Their stability results apply to the case in which the elliptic equation contains a discontinuous but monotone nonlinear term.The associated multivalued equation is of the type −u xx + λβ(u) 0 as, for instance, it arises in the study of the stationary Stefan problem with an absorption term (see [6] and [10]).