Lack of smoothing for bounded solutions of a semilinear parabolic equation

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as $t\to\infty$.

It is well known (see [1,Thm. VI.4.2]) that this problem has a unique classical solution for small T > provided g ∈ C (R n+ ), u ∈ C (Ω) and u = A on ∂Ω. In this paper we study a particular case of problem (0.1) in a radially symmetric setting in B R := {x ∈ R n | |x| < R}, R > , where g is a smooth function of u and ur but u is only Hölder continuous in B R . In our example, the global bounded weak solution emanating from u maintains the singularity of the gradient of u for all t > . Thus, there is no smoothing e ect which one usually expects from a semilinear uniformly parabolic equation.
The equation we will be interested in is the following: forms a stationary solution of (0.2) (for any R > both in B R \ { }, cf. Lemma 5, and -in the weak sense -in B R , see Lemma 6). We will impose several conditions on the initial data u (and refer to (2.14) in Section 2 below for details) that, besides radial symmetry, essentially require that u lies below the stationary solution, but is 'close' to it in a suitable sense. Under these conditions we will be able to show the global existence of solutions that retain the singularity in their gradient throughout the evolution.
By this we mean that uu r ∈ L loc (B R × [ , ∞)) and ∇u ∈ L loc (B R × [ , ∞)), (0.8) We note that Theorem 1 guarantees that the initial and boundary conditions are satis ed.
Concerning the large-time behavior we establish the following: Under the assumptions of Theorem 1, This convergence is uniform in B R and occurs with an exponential rate.
An equation closely related to (0.2) has been studied before in [2,3], see also [4]. It was shown in [2] that interior gradient blow-up may occur for solutions of the problem where m > and f (u) = u, for example. A global continuation after the interior gradient blow-up has been constructed recently in [3] for m = .
For various parabolic equations, solutions with a standing or moving singularity have been investigated by many authors. We shall give some references below. But in these references it is the solution itself that is unbounded while in the present work only the gradient stays unbounded.
For the equation solutions with standing singularities were considered in [5][6][7][8][9][10] for various ranges of m > , m ≠ , and some results on moving singularities for the same equation can be found in [11].
Results on moving singularities for the heat equation were established in [12,13] and for semilinear equations of the form u t = ∆u ± u p , p > , (0. 11) in [14][15][16][17][18][19]. The behaviour of solutions with standing singularity for equation (0.11) with positive sign has been studied in [20,21]. Counterexamples to the regularizing e ect of (0.10) can be found in [22]. There, again, it is the solution itself that is unbounded.
Next we describe the plan of the paper. Due to the gradient singularity that the solutions have at the spatial origin, the notion of classical solvability is restricted to (B R \ { }) × ( , ∞). In Section 1 we therefore begin by establishing a connection between classical solutions in (B R \ { }) × ( , ∞) and weak solutions in B R × ( , ∞). Section 2 will be concerned with the stationary solution u * mentioned in (0.3) (and already use the result of Section 1). At the end of this section, we give a precise formulation of the conditions on u that the theorems require (and that involve the stationary solution).
We will construct the solutions between a super-and a subsolution. As a supersolution we will use u * , nding the subsolution will be the goal of Section 3. To this aim, we will nd a solution v to a (formal) linearization of (0.5) (see Lemma 8) and then ensure that u * − v is a subsolution (Lemma 10). (This is also the source of the restriction on R in the theorems.) The actual construction of solutions takes place in Section 4. We rst restrict the spatial domain to Ωε := B R \ Bε, for the choice of the boundary value on the new boundary ∂Bε × ( , ∞) already relying on u * − v from Section 3. In Section 4.1, we take care of the solvability of this problem. (Classical existence results become applicable after replacing the nonlinearity u r by f (ur), see Lemma 18, and until Lemma 26, we will have derived su cient estimates allowing for removal of f , though still ε-dependent.) Section 4.2 will then be concerned with ε-independent estimates in preparation of a compactness argument leading to the existence of solutions. The key to this part will lie in a comparison principle applied to high powers of ur (see Lemma 27). This is a modi cation of a classical technique which involves |∇u| and originated in [23]. Section 4.3 will contain the passage to the limit ε ↘ (Lemma 32) and deal with (0.5) and (0.7).
In Section 5, nally, we give the proofs of the theorems. By this time, they will only consist in collecting the right lemmata previously proven, and will be accordingly short.

Relation between classical and weak solutions
Of course, every classical solution of (0.5) is also a weak solution of (0.5) -in (B R \ { }) × ( , ∞), which means that the singularity appears on the boundary of the domain. In order to interpret classical solutions in (B R \ { }) × ( , ∞) as weak solutions in B R × ( , ∞), we merely require suitable integrability properties of the derivative near : Lemma 4. Let n ≥ and R > . Assume that a radially symmetric function satis es (0.8), (0.5), and for every T > we have that lim ε→ ε T ε r n− |ur(r, t)|drdt = . (1.12) Then (0.9) holds for every φ ∈ C ∞ c (B R × ( , ∞)).

The stationary solution and conditions on the initial data
In (0.3), we have introduced a stationary solution u * to (0.2). In this section we rst prove that the function from (0.3) actually has this property (see Lemma 5 for the classical, Lemma 6 for the weak sense) and then formulate the conditions on the initial data, which involve relations with u * and whose formulation we therefore had postponed.
Proof. We use radial symmetry and the explicit form of u * to write Lemma 6. Let n ≥ . Then for any R > the function u * de ned in (0.3) is a weak solution of (0.7).
Proof. In order to apply Lemma 4, we only have to check integrability of u * (r)(u * r ) (r) = α r − and u * r (r) = α r − , which is satis ed, and Now and in the following, given any n ∈ N we let Having introduced u * and ν, we are now in a position to give the conditions on initial data that Theorems 1, 2 and 3 have posed.
Remark 7. The shape of the solution from Theorem 1 near the singularity of its gradient can be described more precisely than in (0.6) by saying that (2.14d) continues to hold for t > in the sense that lim sup r↘ |r −n−ν (u * (r) − u(r, t))| < ∞ for all t > .
We will include a proof in the proof of Theorem 1 in Section 5.

Finding a subsolution
In order to construct a subsolution of (0.5) near u * , we rst nd a solution of the (formal) linearization of (0.5) around u * .
where Jν denotes the Bessel function of the rst kind of order ν, solves with u * taken from (0.3).
Proof. Let us recall that the function de ned by χ(r) := Jν(λr), r > , satis es We abbreviate A := − n and B := n− and δ := n − and note that and where we have used that α = As u * is a stationary solution according to Lemma 5 and by Lemma 8 v solves the linearized equation, we conclude

Existence . An approximate problem
Construction of the solution to (0.2) will be based on an appropriately modi ed problem on (B R \ Bε) × ( , ∞).
In preparation of suitable initial data, we rst turn our attention to u .
If we let C ≥ c , this coincides with (4.23).
De nition 12. Now and in all of the following, we let n, C, R, λ, v be as in Lemma 10 and Lemma 11.
Proof. Boundedness of fε and the regularity requirements on u ε ensure applicability of [1, Thm. V.6.2], which yields existence and uniqueness of the solution. Radial symmetry of u ε together with the uniqueness assertion implies radial symmetry of the solution.
Later (in Lemmata 25 and 27) we want to invoke comparison principles for the derivative. In order to make them applicable, we need slightly more regularity than provided by Lemma 18.
As a rst estimate of uε, the following lemma not only a rms boundedness of uε, but also forms the foundation of estimate (4.30) for u.
Proof. Due to (4.25a) and (4.25b), each of the functions w ∈ u * , uε , u * − v satis es fε(wr) = w r in Ωε ×( , ∞) and hence for w ∈ u * , uε we have w t = ∆w + fε(wr)w, We prepare for an estimate of uεr by comparison, rst providing some information on its value on the spatial boundary, beginning with the outer part ∂B R × ( , ∞).
On the inner boundary, we rst establish the sign of uεr.
We now turn our attention to the counterpart of Lemma 22.
The previous lemmata and a rst Bernstein-type comparison of u εr con rm that including fε in (4.26) -although necessary for application of the classical existence theorems -has not altered the equation.

Lemma 25. For every ε > we have sup
Proof

. A priori estimates
Inspired by the reasoning in [2,Sec. 2], which goes back to [23], we will now obtain an ε-independent bound for uεr from a comparison principle applied to, essentially, a large, even power of uεr. Lack of ε-independent control over uεr on the inner boundary (for which we refer to Lemma 24 and which is natural if seen in light of the unbounded derivative of u * near r = ) makes inclusion of a cuto function necessary.
Next we bring Lemma 27 in a more directly applicable form.
As preparation of the compactness argument that will nally establish existence of a solution of (0.5) in (B R \ { }) × ( , ∞), we use classical regularity theory for parabolic PDEs and rely on Lemma 28 as a starting point. The latter convergence statement (4.35) together with Lemma 23 already entails (4.32), whereas (4.33) similarly results from Lemma 28 upon the choice of p = .
Finally, (4.30) and hence (4.31) are obvious for r = and easily obtained from Lemma 20 for r > .
Theorem 1 also includes a uniqueness statement. The following lemma takes care of it. Proof. We observe that according to (4.33) there is c = c (T) such that

Proofs of the theorems
Proof of Theorem 1 and Remark 7. Solvability is ensured by Lemma 32, which by means of (4.30) also ensures that for every t > there are c = c (t) > and c = c (t) > such that ≥ u * (r) − u(r, t) ≥ −v(r, t) ≥ −c r n− Jν(λr) ≥ −c r n− +ν for every r ∈ [ , R].
Proof of Theorem 2. This is the outcome of Lemma 34.