Nonlinear instability of solutions in parabolic and hyperbolic diffusion

We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and determine precise conditions for which linear instability of the steady state solutions implies nonlinear instability. More specifically, we prove that linear instability with an eigenfunction of fixed sign gives rise to nonlinear instability by either exponential growth or finite-time blow-up. We then discuss a few examples to which our main theorem is immediately applicable, including evolution equations with supercritical and exponential nonlinearities.


Introduction
We consider the second-order hyperbolic equation x ∈ R n , t > 0 (1.1) and its parabolic counterpart where L is a second-order elliptic differential operator with smooth, bounded coefficients, f is a nonlinear source, and a(t) is a damping term. An important question in the study of (1.1) and (1.2) is to understand the qualitative behavior of special types of solutions. In this paper, we consider steady-state solutions and determine specific conditions under which they are nonlinearly unstable if it has already been determined that they are linearly unstable. Such criteria are useful in a wide variety of problems, a topic we shall elaborate upon in more detail later. Before stating our main results, we highlight some necessary assumptions. We shall assume throughout that (A1) the elliptic, time-independent partial differential equation Lϕ = f (x, ϕ) has a solution ϕ ∈ C 2 (R n ) (A2) the adjoint linearized operator L * −∂ u f (x, ϕ) has a real, negative eigenvalue, denoted −σ 2 , and corresponding non-negative eigenfunction χ ∈ L 2 (R n ) ∩ L 1 (R n ) (A3) the damping term satisfies a ∈ L 1 loc (0, ∞) (A4) the nonlinearity f (x, u) is C 1 and convex in u.
Formally, our main results state that ϕ is a nonlinearly unstable solution of (1.1) and (1.2) under the assumptions (A1)-(A4). Our results apply to a wide class of initial data, including those for which the initial perturbations are assumed to be nonnegative. In order to prove the main theorems, we will rely on a variant of Kaplan's eigenfunction method [6] which was originally used to study the growth of solutions to quasilinear parabolic equations. Recently, this method was utilized by Strauss and Karageorgis [7] to prove similar theorems for nonlinear PDEs without time-dependent damping terms. The impetus for including sign-changing damping terms arises from a recent paper [2] in which it is shown that steady states of nonlinear wave equations with sign-changing damping coefficients are stable under certain assumptions. In contrast, we will prove that solutions with damping terms satisfying (A3), which allows for coefficients to change sign, are unstable.
To establish our instability result for the hyperbolic case, we show that the energy norm of the perturbed solution must grow exponentially in time. Notice that this is a much stronger result than just instability, as the statement implies that the solution exists in any given neighborhood of ϕ, that it never returns to the region, and that it exits this neighborhood exponentially fast. Under stricter assumptions on the nonlinearity and eigenfunction, the instability can be shown to occur by blow-up. More specifically, if we assume (A5) the functions f (x, ϕ) and ∂ u f (x, ϕ) are bounded (A6) there are C > 0 and p > 1 such that f (x, u) ≥ C|u| p for every (x, u) ∈ R n × R (A7) the product of the eigenfunction and steady state is integrable, i.e. ϕχ ∈ L 1 (R n ), then the energy norm of the perturbed solution must blow-up at some finite time. In the case of the parabolic equation (1.2), we make the additional assumption (A8) the damping coefficient a(t) is positive. This is imposed in order to ensure that the problem remains parabolic and avoid any difficulty stemming from the widely-known ill-posedness of the backward diffusion equation. With this condition, we prove instability in the L ∞ norm by exponential growth under assumptions (A1)-(A4) and instability by blow-up under assumptions (A5)-(A7).
In the special case that the coefficient a(t) is constant, the stability question has already been settled. Specifically, the instability of the steady state ϕ ∈ C 2 of the linearized problems was shown in [7] to imply the instability of ϕ as a solution to the corresponding nonlinear problems with f as in (A4) and a ∈ R, subject to a quadratic condition involving both the value of a and the initial size of the perturbation under consideration. We point out, however, that while the main theorems of [7] required a condition on a ∈ R, our main theorems hold for any choice of a(t) which lies in a suitable L p space. Also, note that dealing with variable coefficients is a non-trivial task, one that can not be settled by a simple change of variables, as no such transformation will render this system into one with constant coefficients.
It is well known that in the presence of the damping term ∂ t u the asymptotic profile of solutions to the linear wave equation is well-behaved and resembles the Gaussian profile of solutions to the linear diffusion equation [11,13]. Moreover, the difference between the two solutions tends to zero as t → ∞ faster than the decay of either solution. This remarkable phenomenon is called the (strong) diffusion phenomenon and was shown to hold for a variety of systems [4,5,12,14]. With the addition of the damping term with time dependent coefficients in the nonlinear problem, the question arises as to whether steady states remain unstable under smooth perturbations for both wave and diffusion equations, and the present study is devoted to addressing this open question.
Regarding the growth of the damping coefficient for the hyperbolic equation, note that a prototype coefficient that satisfies (A3) and the assumptions outlined in both of our main theorems is a(t) = a 0 (1 + t) −α , α > 1.
We point out that coefficients a(t) of growth are not allowed by our assumption (A3). Indeed, it is known (see [17] for example) that the total energy of the linearly damped system with a(t) satisfying (1.4), decays to zero as t → ∞, thus precluding instability. More interestingly, perhaps, our result for the hyperbolic equation allows coefficients that change sign in time. To our knowledge these are the first instability results in this direction, whereas the first stability results for sign-changing systems were reported recently for a nonlinear problem in [2] (based on the earlier work [3]). The authors show in [2,3] that if the damping a(t) is negative for a sufficiently small length of time, then steady solutions remain stable for the system where Ω ⊂ R n is bounded and f (u) is absorbing with subcritical growth (i.e. f (u) = −u|u| p−1 , 1 ≤ p ≤ n n−2 ). In contrast, we will show in the applications section that for specific forcing terms f the system (1.6) with Ω = R n exhibits instability of steady states.
The general interest in studying nonlinear hyperbolic equations in the presence of positive damping often arises from the need to determine the state of a physical system under an energy decreasing force, such as friction. Systems with sign-changing damping are also important in applications as they appear in Aerodynamics, e.g. the nose wheel shimmy of an airplane on which a hydraulic shimmy damper acts [15]; Mesodynamics, as within a laser driven pendulum [1]; Quantum Field Theory, e.g. the Landau instability of Bose condensates [9]; and the macroscopic world, including a well-known model of suspension bridges [10]. We refer the reader to [2] for more references arising in the physics and engineering literature.
The paper is structured as follows. In the next section, we prove the main instability theorems for nonlinear hyperbolic and parabolic equations. Section 3 then contains several lemmas used to prove Theorems 2.1 and 2.2. Finally, in Section 4 we discuss a few examples of well-known problems to which our primary results are immediately applicable.

Main results
In this section, we prove the main instability results regarding solutions of the general elliptic equation We first consider the initial value problem for the previously described hyperbolic equation with x ∈ R n : The steady state ϕ is an exact solution of (2.1) if ψ 0 = ψ 1 ≡ 0. Thus, we are interested in the situation in which ψ 0 and ψ 1 are small in some sense. Our results are subtly different for a ∈ L ∞ and a ∈ L 1 . In short, we find instability for a ∈ L 1 loc , instability by exponential growth for a ∈ L 1 , and instability by either exponential growth or blow-up for a ∈ L ∞ .
Proof. Let w(t, x) = u(t, x) − ϕ(x) and define the function Therefore, G is well-defined as long as the energy remains finite and serves as a lower bound. Hence, the energy grows in time at least as fast as G does, and we may focus on obtaining the growth of G to prove the desired result.
Using the linearity of the left side of the equation in (2.1), we find that w satisfies in the sense of distributions. Due to the convexity of the nonlinearity (A4), we have Since χ is nonnegative, we multiply the inequality by χ(x) and integrate by parts to obtain The assumption (A2) implies Because ∂ t w is continuous with values in the energy space, we see that is continuous. Hence, the differential inequality (2.2) simplifies to become in the sense of distributions. With this, the conclusions of the theorem follow from the lemmas of the next section. More specifically, our assumption on the initial data yields G(0), G ′ (0) > 0. Thus, the first result of Lemma 3.1 proves for all t ∈ (0, T ) and hence the conclusion of part (a). Similarly, conclusions (b) and (d) follow from the results of Lemmas 3.2 and 3.3, respectively, regarding the behavior of functions G(t) that satisfy (2.3).
To prove part (c) we follow [7] with minor modifications, adjusting for the timedependence in the damping term a(t). First assume that T = ∞. Utilizing the additional assumptions (A5) and (A6) and proceeding as before, the equation (2.1) yields As before, we multiply the inequality by the eigenfunction χ, integrate, and use (A2) to find Since χ, χϕ ∈ L 1 (R n ), the last term in the inequality satisfies Put A(t) = R n χ|w + ϕ| p dx and B(t) = R n χ|w + ϕ| dx. Then, since we know from (2.4) that G stays positive, the differential inequality (2.5) implies for t chosen large enough. Using Holder's inequality we see that A(t) dominates B(t) for large t since for any p ∈ (1, ∞) Additionally, notice that By the conclusion of part (b) of the theorem, G grows exponentially fast. Therefore, for t large we find that B(t) must also grow exponentially fast, as it is lower bounded by G. Using these facts in (2.6), we find By conclusion (a), we know G ′ is positive, and since a ∈ L ∞ , it follows from this inequality that for t large enough. From Lemma 3.4, we then conclude that T < ∞.
Remark 2.1. The assumptions on the initial data can be lessened in the case of a ∈ L ∞ (0, T ). More specifically, the conclusion of the theorem remains valid under the assumption The proof of this corollary follows from adapting the above argument to the methods of [7], in which a similar condition is imposed on the constant damping coefficient.
Next, we present the instability analogue of Theorem 2.1 to the parabolic problem on R n : Unlike the hyperbolic case our main theorem proves instability in the L ∞ norm, rather than the energy norm. In either situation, these norms are natural to the respective local existence theory. In short, we find instability for a ∈ L 1 loc and instability by either exponential growth or blow-up for a ∈ L ∞ .

Theorem 2.2. (Parabolic equation)
Assume (A1)-(A4) and (A8) with initial data ψ 0 ∈ L ∞ (R n ) satisfying Let T ∈ (0, ∞] be given, and let u be a solution of (2.8) on [0, T ) such that u − ϕ is C 1 and bounded for each t. Then, we have the following: a. There is C 0 > 0 such that for all t ∈ (0, T ), Proof. As in the hyperbolic case we define for which we have the estimate So the blow-up of G will determine the blow-up of w, and hence the instability result. We apply the argument of Theorem 2.1 and by imposing the convexity assumption on f we see that G must satisfy the analogue of (2.3), namely Applying Lemma 3.5 gives the validity of the first two conclusions of the theorem. To obtain the last part of the theorem we use the same estimates that yielded (2.7); the parabolic counterpart is now given by By Lemma 3.5(i) we have that G is positive, hence we can directly apply Lemma 3.6 to obtain that G blows up in finite time.

Lemmas
In this section, we state and prove the lemmas used in the proof of Theorems 2.1 and 2.2. We will first consider the differential inequalities which arise from our study of the hyperbolic problem (2.1), namely and for p ∈ (1, ∞).
Proof. To prove the first lemma, we shall take advantage of the regularity of Y (t) and utilize a continuity argument. LetT < T be given so thatT < ∞. Define We have T 1 ≤T < T ; moreover, T 1 > 0 since Y ′ (0) > 0 by hypothesis. Hence, by continuity there exists T 0 ∈ (0, T 1 ) such that Since Y satisfies (3.1) on (0, T ) in the sense of distributions, it will also satisfy it on (T 0 , T 1 ), hence we have Taking θ to be an approximation of e I(s) χ (T 0 ,T 1 ) (s), where χ (a,b) represents the characteristic function on (a, b), we deduce By the continuity of Y ′ (t) it must be the case that T 1 =T , so Y ′ (t) > 0 for all t ∈ [0,T ). WithT < T arbitrary we have that Y ′ (t) > 0 on [0, T ), which is the desired conclusion.
Next, we prove the exponential growth of solutions under similar conditions.
Put a ∞ := a ∞ and let λ ± be the roots of the characteristic equation Then, which implies Z(t) ≥ Z(0)e λ + t and by the definition of Z, Finally, this generates a lower bound for Y , namely Since λ − < 0, we have Z(0) = Y ′ (0) − λ − Y (0) > 0 by assumption. Because λ + > 0, the exponential growth of Y follows. Now we show that a similar result for a ∈ L 1 also holds, but with a different method of proof. Lemma 3.3. Let T ∈ (0, ∞], a ∈ L 1 (0, T ), and b > 0 be given. Suppose Y ∈ C 1 satisfies (3.1) on [0, T ) in the sense of distributions with given initial conditions Y (0), Y ′ (0) > 0. Then, there exists C, λ > 0 such that Proof. Let a ∈ L 1 (0, T ) and b > 0 be given. Choose and notice that T 0 > 0. Thus, for any t ∈ [0, T 0 ], we can proceed as in the proof of Lemma 3.1 to express (3.1) as where I(s) = s 0 a(τ ) dτ . As before, we take θ to be an approximation of the integrating factor multiplied by the characteristic function on (0, t) and the inequality becomes Since a is integrable, we find and after an integration Reordering terms, this becomes By definition, λ 2 ≥ 1 2 e − a 1 b, thus the first term on the right side of (3.3) is nonnegative. Similarly, the second term is nonnegative by hypothesis. Finally, call the last term L. Since e x > 5 2 x for all x ∈ R and Y (0) ≥ 0 we find
We state the next lemma to justify the proof of part (c) in Theorem 2.1 and refer the reader to Proposition 3.1 in [16] for its proof.
Lemma 3.4. Let A, B > 0, and p > 1 be given. Let Y ∈ C 1 be a solution of (3.2) on the interval [0, T ) in the sense of distributions with given initial conditions Finally, we turn to the lemmas for the parabolic case and consider the differential inequalities and for p ∈ (1, ∞), which arise in the proof of Theorem 2.2.
Lemma 3.5. Let T ∈ (0, ∞], a ∈ L 1 loc (0, T ) be positive, and b > 0 be given. Suppose Y ∈ C 1 satisfies (3.5) on [0, T ) with given initial condition Y (0) > 0. Then Proof. We proceed as in the proofs of Lemma 3.1 and Lemma 3.2 using a continuity argument and define Note that on [0, T 0 ) we have by (3.5) Hence, the positivity of a implies Y ′ (t) > 0 as well. This yields Y (t) > Y (0) for all t ∈ (0, T 0 ), whence T 0 = T , and the first conclusion of the theorem holds. For the second part of the lemma, we use the L ∞ bound on a and the positivity of Y to obtain from (3.5) the inequality As in the proof of Lemma 3.2, the exponential growth of Y on [0, T ) follows as an immediate consequence.
Our final lemma of the section establishes the blow-up result for (3.6).
Proof. Assume T = ∞. As in the previous lemma, a continuity argument easily establishes Y (t) > Y (0) and Y ′ (t) > 0 for t ∈ [0, T ). Using this (3.6) implies The inequality can be rearranged in separable form which after integration and algebraic manipulations gives .
Since there exists a time T ∞ < ∞ such that we conclude that Y blows up as t approaches T ∞ .
Remark 3.1. Lemma 3.6 holds under a different assumption that the coefficient a In fact only the milder condition is necessary. This last assumption, however, would be more difficult to verify in the context of Theorem 2.2, since it would involve computing both the eigenfunction χ and the eigenvalue −σ 2 on which C depends. We prefer the condition that a ∈ L ∞ for part (b) of Theorem 2.2, since this coincides with the assumption needed for part (a) and it is also very easy to verify in applications.

Applications of Theorems 2.1 and 2.2
Our results are immediately applicable in many specific situations, including those described by Strauss and Karageorgis in [7]. Namely, for steady state solutions that satisfy the following equations, the instability via exponential growth and finite time blow-up results of Theorems 2.1 and 2.2 are valid.
The previous result may be obtained in the more general settings of the heat and wave equations with steady state solutions satisfying −∆ϕ + V (x)ϕ = f (ϕ), x ∈ R n where f is a C 1 convex function with f (0) = f ′ (0) = 0, not identically zero, only if one can show existence of such positive, C 2 steady state solutions that decay at infinity (see Assumptions of Theorem 3.1 in [7]).

Exponential nonlinearity in 2D:
One may also transfer instability from the linearized systems to the nonlinear problems ∂ tt u − ∆u + a(t)∂ t u = e u , a(t)∂ t u − ∆u = e u . (4.4) Note that f (y) = e y is a convex function and the following hold (see section 4 in [7]): • f (ϕ) is bounded for all ϕ positive solutions of the steady state equation −∆ϕ = e ϕ , e ϕ ∈ L 1 (R 2 ) • The linearized operator v → Lv = −∆v − e ϕ(x) v has the first eigenvalue negative and the first eigenfunction positive.
a(t)∂ t u − ∆u = |u| p , ∂ tt u + a(t)∂ t u − ∆u = |u| p , where a(t) satisfies the assumptions given in Theorems 2.1 and 2.2, and instability occurs as given by the aforementioned theorems.
If p ≥ p c , then ϕ is a stable solution of the linearized equation, i.e. the linearized operator has no negative spectrum.
To our knowledge, the instability results of Theorem 4.3 are new for both the parabolic and hyperbolic equations. Again, for a more detailed discussion of the case when a is constant we refer the reader to [7].