Stability of degenerate parabolic Cauchy problems

We prove that solutions to Cauchy problems related to the $p$-parabolic equations are stable with respect to the nonlinearity exponent $p$. More specifically, solutions with a fixed initial trace converge in an $L^q$-space to a solution of the limit problem as $p>2$ varies.


Introduction
We study the stability of solutions to the degenerate parabolic equation ∂ t u − div(|∇u| p−2 ∇u) = 0 (1.1) with respect to perturbations in the nonlinearity power p. The main issue we address is whether solutions converge in some sense to the solution of the limit problem as the nonlinearity power p varies. In applications, parameters like p are often only known approximately, for instance from experiments. Thus it is natural to ask whether solutions are sensitive to small variations in such parameters or not. The stability of Dirichlet problems on bounded domains for (1.1) is detailed in [KP10]. See also [Lind87,LM98,Lind93] for elliptic equations similar to the p-Laplacian, [Luk] for the porous medium and fast diffusion equations, and [FHKM] for parabolic quasiminimizers. Here we focus on the Cauchy problem ∂ t u − div(|∇u| p−2 ∇u) = 0, in R n × (0, ∞) u(x, 0) = ν with p > 2. The initial trace ν can be a positive measure with compact support and finite total mass, and the interpretation of the initial condition is that the initial values is attained in the sense of distributions. See Section 2 below for the details. For the existence theory of this initial value problem, we refer to [DH89].
An important example of solutions to the Cauchy problems is given by the celebrated Barenblatt solutions. These functions are the nonlinear counterparts of the fundamental solution, and they are defined for positive times t > 0 by the formula where λ = n(p − 2) + p. The constant C is at our disposal. We choose the value for which R n B p (x, t) dx = 1 for all t > 0.
With this normalization, one easily checks that lim t→0 B p (·, t) = δ 0 , in the sense of distributions, where δ 0 is Dirac's delta at the origin of R n . Further, the function B p is the unique solution to the Cauchy problem with this initial trace, see [KV88].
Our main result states that solutions and their gradients converge to a limit function in an L q -space, locally in space and up to the initial time, as p varies. The limit function turns out to be a weak solution to the equation involving the limit exponent, and it also has the correct initial trace. In particular, since the Barenblatt solutions are the unique solutions to the respective Cauchy problems, our stability theorem contains the fact that in the above sense. The admissible exponents in the norms are sharp in the sense that for larger exponents, the norms of the Barenblatt solution and its gradient are no longer finite. For general initial measures, uniqueness for the Cauchy problem remains open, and we only get a subsequence converging to some solution of the limit problem. If the initial trace is an L 1 function, then the solution to the limit problem is unique, and the whole sequence converges. The proof of our stability result is based on deriving suitable estimates for solutions and their gradients. We use the truncation device commonly employed in measure data problems for this purpose, see e.g. [BDGO97]. This works well, since the mass of the initial data has the same role in the estimates as the mass of a right hand side source term. Convergence of the solutions is established by employing the ideas of [KP10] and a localization argument. Here we encounter a difficulty: changing the growth exponent p changes the space to which solutions belong. To deal with this, we need to use a local Reverse Hölder inequality for the gradient. Such a result can be found in [KL00].
The paper is organized as follows. In Section 2, we present the necessary preparatory material, including the parabolic Sobolev spaces to which weak solutions belong, and a rigorous description of what is meant by the Cauchy problem in this context. In Section 3, we prove a result concerning the propagation properties of solutions. These properties are then employed in Section 4 in proving the estimates necessary for our stability result. Finally, in Section 5 we prove the main theorem about stability.

Preliminaries
We denote by L p (t 1 , t 2 ; W 1,p (Ω)), where t 1 < t 2 , the parabolic Sobolev space. This space consists of measurable functions u such that for almost every t, t 1 < t < t 2 , the function x → u(x, t) belongs to W 1,p (Ω) and the norm The definitions for L p loc (t 1 , t 2 ; W 1,p loc (Ω)) and L p (t 1 , t 2 ; W 1,p 0 (Ω)) are analogous. The space C((t 1 , t 2 ); L q (Ω)), q = 1, 2, comprises of all the functions u such that for t 1 < s, t < t 2 , we have Ω |u(x, t) − u(x, s)| q dx → 0 as s → t. The notation U ⋐ Ω means that U is a bounded subset of Ω and the closure of U belongs to Ω. We also denote U ε = U × (ε, ∞). We study the Cauchy problem div |∇u| p−2 ∇u = ∂u ∂t , in R n × (0, ∞), u(·, 0) = ν, where ν is a compactly supported positive Radon measure. For simplicity, this measure is fixed throughout the paper. To be more precise, the solution u ≥ 0 satisfies the initial condition in the sense of distributions, meaning that for all ϕ ∈ C ∞ 0 (R n ). Further, u is a weak solution for positive times in the sense of the following definition.
Our assumptions imply that the Cauchy problem has a solution, see [DH89] and [KP09]. These solutions are constructed by approximating the initial trace by regular functions, proving uniform estimates, and then passing to the limit by compactness arguments.
Remark 2.5. By a solution to the Cauchy problem (2.1) we mean any solution obtained as a limit of approximating the initial trace ν via standard mollifiers. The reason for this is that we want to choose the approximations so that their support does not expand much.
The above definition does not include a time derivative of u in any sense. Nevertheless, we would like to employ test functions depending on u, and thus the time derivative of u inevitably appears. We deal with this defect by using the convolution u σ (x, t) = 1 σ t ε e (s−t)/σ u(x, s) ds, (2.6) defined for t ≥ ε > 0. This is expedient for our purpose; see for example [Nau84], [BDGO97], and [KL06]. In the above, ε > 0 is a Lebesgue instant of u. The reason for using a positive number ε instead of zero is that solutions are not a priori assumed to be integrable up to the initial time.
The convolution (2.6) has the following properties. and To derive the necessary estimates, we need the equation satisfied by the mollified solution u σ . By using (2.4), we may write Then we multiply the above inequality by e −s/σ /σ, integrate over [0, T − ε] with respect to s, change the order of integration using . . ds dt, and finally perform a change of variables s new = s + ε, t new = t + ε, and for simplicity denote the new variables also with s and t . We end up with (2.8) It is often convenient to write the last two terms on the right as where we integrated by parts in time and used the fact that u σ (x, ε) vanishes by the properties of the mollifier.

Propagation properties
In this section, we prove a finite propagation result for the Cauchy problem. More specifically, if the initial trace has compact support, then this property is preserved in the time evolution, meaning that x → u(x, t) also has compact support for positive times t. This is in principle well known, but it seems hard to find a convenient reference covering the case when the initial trace is a measure. For stronger initial conditions, see for example Theorem 3.1 in [KP09].
Here we use a comparison with an explicit solution and adapt the arguments for the porous medium equation in [Vaz06], see in particular Lemma 14.5 and Proposition 14.24. We start by extracting a family of barrier functions from the Barenblatt solution is also a weak solution, since both the terms in the p-parabolic equation change sign. With the choice C = 0 and some simplifications, we get a weak solution (3.1) The first result concerns finite propagation for bounded initial data.
Theorem 3.2. Let u be the solution to the Cauchy problem where u 0 ≥ 0 and ||u 0 || ∞ = H < ∞. Assume that there is a point x 0 ∈ R n and a radius R > 0 such that Then there exists a nonnegative function R(t) such that Remark 3.3. It can be shown that the function R(t) is nonincreasing, but we do not need this fact here. In Theorems 3.2 and 3.5, it can happen that R(t) = 0 after some time instant.
Proof. By the comparison principle, we have 0 ≤ u ≤ H. Pick a point We derive an estimate for the time T up to which u(x 1 , T ) = 0 by comparing u with the barrier function v given by Choose now T small enough so that equality holds in (3.4). Then The next step is to generalize the above result to the case of initial data with finite mass. For this purpose, we need the so-called L 1 − L ∞ regularizing effect, i.e. the estimate Here λ = n(p − 2) + p, α = n/λ, and σ = p/λ, see Theorem 2.1 in [DiB93] on page 318. The proof is an iteration of the estimate in Theorem 3.2 with the help of the regularizing effect.
Theorem 3.5. Assume that ν is a positive measure with ν(R n ) < ∞, and that there is a point x 0 ∈ R n , a radius R > 0, and a number ε > 0 such that ν(B(x 0 , R + ε)) = 0.
Then there is a solution u to the Cauchy problem Then there exists a nonnegative function R(t) such that and u(x, t) = 0 for all x ∈ B(x 0 , R(t)) where γ = 1 p 1 − n(p − 2) n(p − 2) + p and c(n, p, ν) = c(n, p)ν(R n ) σ(p−2)/p .
Proof. We derive the estimate for the case where the initial trace is a bounded function u 0 such that The general case then follows by mollifying the initial trace ν so that the above assumption holds for the approximations, and passing to the limit. Indeed, the estimate depends on the initial trace only via its total mass, so the result holds also for the limit function. Fix a small t > 0 and let t k = 2 −k t for k = 0, 1, 2, . . .. Then t k−1 = 2t k and t k−1 − t k = t k . Denote H k = ||u(·, t k )|| ∞ ; then by the regularizing effect. Further, we have We combine the estimate of Theorem 3.2 over the time interval (t k+1 , t k ) and the last two display formulas to get Iteration of the above estimate leads to The series is convergent since γ is positive, and the proof is complete.
Remark 3.6. We will use Theorem 3.5 in the following way: given a time T and an initial trace ν with compact support, we can find a bounded open set U such that the support of u(·, t) is contained in U for all 0 < t ≤ T . To see this, note that we may choose the point x 0 in Theorem 3.5 to be arbitrarily far away from the support of ν.

A priori estimates
In this section, we derive some estimates we will employ for our stability result. More specifically, we estimate certain L q -norms of solutions and their gradients locally in space, up to the initial time. We begin with a simple lemma for passing up to the initial time in our estimates. Proof. Since u is a local weak solution for positive times, it is also continuous in time with values in L 2 loc (R n ) on (0, ∞). Thus x → u(x, ε) is a locally integrable function for every ε > 0, and we may identify it with the measure defined by Observe that we need the continuity in time only for strictly positive times. Now the fact that ν is the initial trace of u means exactly that ν ε → ν in the sense of weak convergence of measures. Thus by the properties of weak convergence of measures.  Proof. We test the regularized equation with ϕ = u j = min(u, j). This is an admissible test function, since u has a 'finite speed of propagation' i.e. for a given time T , there is U such that u(·, t) is supported on U, see Remark 3.6. We get We need to eliminate the time derivative in the first term on the left. To accomplish this, we note that ∂u σ ∂t u j = ∂u σ ∂t min(u σ , j) + ∂u σ ∂t (min(u, j) − min(u σ , j)) ≥ ∂u σ ∂t min(u σ , j), so that this term is estimated from below by The time derivative has been eliminated, so we may pass to the limit σ → 0. We take this limit and get We then use the fact that u j ≤ j Then pass to the limit ε → 0; the right hand side is bounded by cjν(U ) by Lemma 4.1, so we get the desired quantity to the right hand side by a trivial estimate. The proof is then completed by replacing T in the above by τ chosen so that . This leads to an estimate for the supremum in terms of the right hand side in the claim.
We need the following well known Sobolev type inequality.
Next we show that solutions to the Cauchy problem, as well as their gradients, are integrable to certain powers. The proof is based on estimating the decay of certain level sets by applying the previous two lemmas. This is optimal as can be seen from the Barenblatt solution.
The essential point is that the constant on the right can be chosen to be independent of u.
Above the constant remains bounded when p → p 0 > 2.
Proof. Since T and ν are given, we can choose open U ⋐ R n such that u(·, t) is compactly supported in U for each t ∈ [0, T ], see Remark 3.6. Let us define the sets The reason for introducing a positive ε > 0 here is that we only know the integrability of the solution u and its gradient for strictly positive times. The aim is to derive estimates independent of ε on the time interval (ε, T ), and then pass to the limit ε → 0. This is possible since the right hand side in the estimate of Lemma 4.2 is independent of ε.
Recall the notation u 2j = min(u, 2j). We have with a constant independent of ε.
With (4.6), the estimates follow by essentially the same computations as in the proof of Lemma 3.14 in [KL05]. By (4.6), we get The sum on the last line is finite by the choice of q. We let ε → 0 to conclude the desired estimate for u.
Next we estimate the gradient. We use Hölder's inequality, (4.6) and Lemma 4.2 to obtain We apply Lemma 4.2 once more to estimate the last integral by C2 j−1 . This gives the bound for the second term. Here the sum converges since 1−p−p/n+q+q/n < 0. Letting ε → 0 completes the proof.

Stability
Our stability result now follows from the estimates of the previous section by arguments similar to those in [KP10].
We work with a sequence of exponents (p i ) such that lim i→∞ p i = p > 2.
By the convergence, we are free to assume that p i > 2, and that all the exponents belong to a compact subinterval [p − , p + ]of (2, ∞).
Theorem 5.1. Let (p i ) be a sequence such that p i → p > 2, (u i ) the corresponding solutions to (2.1) (see Remark 2.5), T > 0 and U ⋐ R n .Then there exists a subsequence still denoted by (u i ) and a function u ∈ L q (0, T ; W 1,q (U)), q < p − 1 + 1/(n + 1), such that and ∇u i → ∇u weakly in L q (U T ), Proof. Since q > 1, weak convergence follows from the reflexivity of L q and the uniform L q -bounds of Theorem 4.5. For the norm convergence of the solutions, we estimate the time derivative of u and appeal to the parabolic version of Rellich's theorem by Simon [Sim87], see also page 106 of Showalter's monograph [Sho97]. Denote 1/q ′ + 1/q = 1. For all ϕ ∈ C ∞ 0 (U T ), we have for any φ ∈ C ∞ 0 (U T ) ∩ L q (0, T ; W 1,q 0 (U)). Since (p − 1)q ′ < q, it follows that U T |∇u i | (p−1)q ′ dx dt 1/q ′ < C. Thus, by the density of smooth functions in L q (0, T ; W 1,q 0 (U)), the time derivative of u i is bounded in the dual space of L q (t 1 , t 2 ; W 1,q 0 (Ω ′ )). Now the theorem mentioned above implies the claim.
The previous theorem is not sufficient for our stability result, since the weak convergence of the gradients is not enough to identify the weak limit of the nonlinear quantity |∇u i | p i −2 ∇u i . This is rectified in the following theorem by proving the pointwise convergence of the gradients.
Theorem 5.2. Let (p i ), (u i ), q, U be as in Theorem 5.1. There exists a subsequence (u i ) and a function u ∈ L q (0, T ; W 1,q (U)) such that Proof. By Remark 3.6, we can again assume that spt u(·, t) ⊂ U. We can focus our attention to the convergence of the gradients in L q (U T ). To establish this, let u j and u k be two solutions in the sequence. Since both u j and u k satisfy the mollified equation (2.4), by subtracting, we We use the test function Also observe that p j , p k are close enough to p, both the functions actually belong to a higher parabolic Sobolev space with power p + δ with some δ > 0 by the higher integrability result in [KL00].
We estimate the first term on the left hand side of (5.3). A substitution of the test function and the properties of the convolution imply This estimate is free of the time derivatives of the functions u j and u k , which is essential for us in the passage to the limit with σ. Now, letting σ → 0, we conclude that Observe that the first term on the right hand side is nonpositive. Thus (5.4) We divide the left hand side in three parts as First, we concentrate on I 2 and I 3 . A straightforward calculation shows that where ζ ∈ R n and a, b ≥ 0. If |ζ| ≥ 1, then and if |ζ| ≤ 1, then This leads to for every ζ ∈ R n and a, ≥ 0. Next we apply (5.6) with ζ = ∇u j , a = p j − 2, and b = p − 2. This implies The integral on the right hand side is uniformly bounded for small enough γ > 0 by Theorem 4.5. Consequently, I 2 → 0, as j, k → ∞. A similar reasoning implies that I 3 tends to zero as j, k → ∞. From the elementary inequality (5.4) as well as (5.5) we conclude that The right hand side can be made arbitrary small by choosing j and k large enough, and a suitable ε since the last integral converges for almost every ε (the choice of ε can be done independent of j, k). This shows that (∇u i ) is a Cauchy sequence in L p (U ε,T ), and thus it converges. We can choose a subsequence such that ∇u i → ∇u a.e. in U ε,T as i → ∞, and further by diagonalizing with respect to ε we can pass to subsequence such that ∇u i → ∇u a.e. in U T as i → ∞. This and the uniform estimate in Theorem 4.5 implies that as i → ∞, for q < p − 1 + 1/(n + 1). To see this, pickq such that q <q < p−1+1/(n+1). By the pointwise convergence of the gradients established above, we also have convergence in measure. For any λ > 0, we have by Hölder's inequality. By the convergence in measure and the bound in Lq, we get that lim sup j→∞ ||∇(u − u j )|| L q (U T ) ≤ cλ.
Since λ was arbitrary, the claim follows from this.
Theorem 5.7. Let (p i ), (u i ), q, U, and u be as in Theorem 5.1. Then there is a subsequence (u i )and a function u such that u i → u in L q (0, T ; W 1,q (U)).
If the initial trace ν is such that the solution to the limiting Cauchy problem is unique, then the whole sequence converges to the unique solution.
Remark 5.8. The uniqueness of solutions to the Cauchy problem is open for general initial measures ν. However, uniqueness holds if ν is an L 1 function, see [DH89]. Further, uniqueness is known for the special case of Dirac's delta, and then the unique solution is the Barenblatt solution, see [KV88]. Hence (1.2) follows from this theorem. as t 1 → 0. As then i → ∞, we obtain t 2 +k t 2 R n u i ϕ dx dt → t 2 +k t 2 R n uϕ dx dt due to Theorem 5.1. Finally, by passing to zero with k, it follows that since the weak solution u belongs to C((0, T ), L 2 loc (R n )). Observe that we only use the continuity on an open interval (0, T ).
Consider next the right hand side of (5.9). Let first h → 0 and t 1 → 0 in this order. The uniform integrability estimate in Theorem 4.2 implies t 2 +k 0 R n |∇u i | p−2 ∇u i · (χ 0,k 0,t 2 ∇ϕ) dx dt (5.10) The right hand side tends to zero, as t 2 → 0, and we have shown that u takes the right initial values. It remains to prove the claim about the situation when we have uniqueness. We argue by contradiction. Let u be the unique solution to the limit problem. If the whole sequence does not converge to u, there are indices i k , k = 1, 2, 3, . . ., and a number ε > 0 such that lim k→∞ ||u − u i k || L q (0,T ;W 1,q (U )) ≥ ε.
The first part of the theorem applies to this subsequence, and we get lim inf k→∞ ||u − u i k || L q (0,T ;W 1,q (U )) = 0 by uniqueness, which is a contradiction. The proof is complete.