Blow up of solutions of semilinear heat equations in non radial domains of $\mathbb R^2$

We consider the semilinear heat equation \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll}v_t-\Delta v= |v|^{p-1}v&\mbox{in}\Omega\times (0,T)\\ v=0&\mbox{on}\partial \Omega\times (0,T)\\ v(0)=v_0&\mbox{in}\Omega \end{array}\right.\tag{$\mathcal P_p$} \end{equation} where $p>1$, $\Omega$ is a smooth bounded domain of $\mathbb R^2$, $T\in (0,+\infty]$ and $v_0$ belongs to a suitable space. We give general conditions for a family $u_p$ of sign-changing stationary solutions of \eqref{problemAbstract}, under which the solution of \eqref{problemAbstract} with initial value $v_0=\lambda u_p$ blows up in finite time if $|\lambda-1|>0$ is sufficiently small and $p$ is sufficiently large. Since for $\lambda=1$ the solution is global, this shows that, in general, the set of the initial conditions for which the solution is global is not star-shaped with respect to the origin. In previous paper by Dickstein, Pacella and Sciunzi this phenomenon has already been observed in the case when the domain is a ball and the sign changing stationary solution is radially symmetric. Our conditions are more general and we provide examples of stationary solutions $u_p$ which are not radial and exhibit the same behavior.

It is well known that the initial value problem (1.1) is locally well-posed in C 0 (Ω) and admits both nontrivial global solutions and blow-up solutions. Denoting with T v 0 the maximal existence time of the solution of (1.1), we define the set of initial conditions for which the corresponding solution is global, i.e. and its complementary set of initial conditions for which the corresponding solution blowsup in finite time: For a fixed w ∈ C 0 (Ω), w = 0 and v 0 = λw, λ ∈ R, it is known that if |λ| is small enough then v 0 ∈ G, while if |λ| is sufficiently large then v 0 ∈ B.
Moreover, for any N ≥ 1, considering nonnegative initial data, it can be easily proved that the set G + = {v 0 ∈ G, v 0 ≥ 0} is star-shaped with respect to 0 (indeed it is convex, see [8]). On the contrary when the initial condition changes sign G may not be star-shaped.
Indeed for N ≥ 3, in [1] and [7] it has been shown that there exists p * < p S , with p S = N +2 N −2 , such that ∀ p ∈ (p * , p S ) the elliptic problem admits a sign a changing solution u p for which there exists ǫ > 0 such that if 0 < |1−λ| < ǫ then λu p ∈ B. More precisely this result has been proved first in [1] when Ω is the unit ball and u p is any sign-changing radial solution of (1.2), and then in [7] for general domains Ω and sign-changing solutions u p of (1.2) (assuming w.l.g. that u + p ∞ = u p ∞ ), satisfying the following conditions as p → p * , where S is the best Sobolev constant for the embedding of H 1 0 (Ω) into L 2 * (Ω). When N = 1 such a result does not hold since it is easy to see that for any sign changing solution u p of (1.2), λu p ∈ G for |λ| < 1 and λu p ∈ B for |λ| > 1.
The case N = 2 was left open in the papers [1] and [7], mainly because there is not a critical Sobolev exponent in this dimension so that the conditions and results of these papers are meaningless. Recently inspired by [5], [6], Dickstein, Pacella and Sciunzi in [4] succeeded to prove a blow up theorem similar to the one in [1], considering radial sign changing stationary solutions u p of (1.2) in the unit ball for large exponents p. Indeed, the condition p → +∞ in dimension N = 2 can be considered to be the natural extension of the condition p → p S for N ≥ 3. In this paper we consider again the case N = 2 but the bounded domain Ω is not necessarily a ball. We deal with sign-changing solutions u p of (1.2) with the following two properties: where, for R > 0, Our main result is the following Theorem 1.1. Let N = 2 and u p be a family of sign-changing solutions of problem (1.2) satisfying (A) and (B). Then, up to a subsequence, there exists p * > 1 such that for p > p * there exists ǫ = ǫ(p) > 0 such that if 0 < |1 − λ| < ǫ, then A few comments on conditions (A) and (B) are needed. The first one is an estimate of the asymptotic behavior, as p → +∞, of the energy of the solutions. It is satisfied by different kinds of sign changing solutions (see [2], [3], [5]), in particular by the radial ones in the ball (see [4], [6]). The condition (B) is more peculiar and essentially concerns the asymptotic behavior of |u p (y)| p−1 for points y which are not too close to x + p ; note that p|u p (y)| p−1 can also be divergent since lim inf p→+∞ u p ∞ ≥ 1. It is satisfied, in particular, by sign changing radial solutions u p,K of (1.2) having any fixed number K of nodal regions (see Section 7 for details). But it also holds for a class of sign changing solutions in more general domains as we show in the next theorem. Putting together Theorem 1.1 and Theorem 1.2 one gets the extension of the blow-up result of [4] to other symmetric domains. Note that this in particular includes the case of sign changing radial solutions in a ball, providing so a different proof of the result of [4] which strongly relied on the radial symmetry.
The proof of Theorem 1.1 follows the same strategy as the analogous results in [1,7,4] being a consequence of the following proposition, which is a particular case of [1, Theorem 2.3] Proposition 1.4. Let u be a sign changing solution of (1.2) and let ϕ 1 be a positive eigenvector of the self-adjoint operator L given by Lϕ = −∆ϕ−p|u| p−1 ϕ, for ϕ ∈ H 2 (Ω)∩ H 1 0 (Ω). Assume that Then there exists ǫ > 0 such that if 0 < |1 − λ| < ǫ, then λu ∈ B.
More precisely we will show that under the assumptions (A) and (B) condition (1.5) is satisfied for p large (see Theorem 4.1 in the following). This proof is based on rescaling arguments about the maximum point of u p : using the properties of the solution of the limit problem, we analyze the asymptotic behavior as p → +∞ of the rescaled solutions and of the rescaled first eigenfunction of the linearized operator at u p . In this analysis the assumption (B) is crucial.
To get Theorem 1.2 we prove a more general result which shows that condition (B) holds for a quite general class of solutions u p of (1.2) (see Lemma 5.1 and Theorem 5.3).
The paper is organized as follows.
In Section 2 we collect some properties satisfied by any family of solutions u p under condition (A) and we give a characterization of condition (B). In Section 3 we carry out an asymptotic spectral analysis under assumption (A) and (B), studying the asymptotic behavior of the first eigenvalue and of the first eigenfunction of the linearized operator at u p . Section 4 is devoted to the proof of Theorem 1.1 via rescaling arguments. In Section 5 we find a sufficient condition which ensures the validity of property (B) (see Lemma 5.1) and we select a general class of solutions to (1.2) which satisfies this sufficient condition (see Theorem 5.3). In Section 6 we prove Theorem 1.2.
Finally in Section 7 we show that also the sign-changing radial solutions in the unit ball satisfy the assumptions of Theorem 1.2.

Preliminary results
We fix some notation. For a given family (u p ) of sign-changing stationary solutions of (1.1) we denote by In the next two lemmas we collect some useful properties holding under condition (A).
Lemma 2.1. Let (u p ) be a family of solutions to (1.2) and assume that (A) holds. Then Proof. It is well known, see for instance [3, Lemma 2.1] or [5].
Lemma 2.2. Let (u p ) be a family of solutions to (1.2) and assume that (A) holds. Then, up to a subsequence, the rescaled function is the solution of the Liouville problem Proof. It is well known, see for instance [

3, Proposition 2.2 & Corollary 2.4]
Observe that, under condition (A), by (2.7), for any R > 0 there exists p R > 1 such that the set {y ∈ Ω, |y − x + p | > Rµ + p } = ∅ for p ≥ p R . As a consequence for any R > 0 the value Next we give a characterization of condition (B): Assume that u p is a family of sign-changing solutions to (1.2) which satisfies condition (A).
Then for any R > 0 there exists Proof. For R > 0 and p > 1 define the set In order to prove the equivalence it is enough to show that for any R > 0 there exists 9) for p ≥ p R and the union is disjoint. Indeed (2.9) implies that and so the conclusion. By definition x + p ∈ N + p , moreover by (2.6) and (2.7) → +∞ as p → +∞ and so for any R > 0 there exists p R > 1 such Hence (2.9) follows from the fact that and the union is disjoint. 3. Asymptotic spectral analysis 3.1. Linearization of the limit problem. In [4] the linearization at U of the Liouville problem (2.5) has been studied. In the following we recall the main results.

3.2.
Linearization of the Lane-Emden problem. In this section we consider the linearization of the Lane-Emden problem and study its connections with the linearization L * of the Liouville problem.
We define the linearized operator at u p of the Lane-Emden problem in Ω . Let λ 1,p and ϕ 1,p be respectively the first eigenvalue and the first eigenfunction (normalized in L 2 ) of the operator L p . It is well known that λ 1,p < 0, ∀ p > 1 and that ϕ 1,p ≥ 0, moreover ϕ 1,p L 2 (Ω) = 1.
Proof. For R > 0 we divide the integral in the following way since, up to a subsequence, v + p → U in C 1 loc (R 2 ) as p → +∞ (see Lemma 2.2) and so V p → e U uniformly in B R (0), up to a subsequence. On the other hand Following [4] we estimate Last we estimate  Proof. We divide the proof into two steps.
and so the conclusion follows by Lemma 3.2.
Step 2. We show that, up to a subsequence, for ǫ > 0 λ 1,p ≤ λ * 1 + ǫ for p sufficiently large. The proof is similar to the one in [4], we repeat it for completeness. For R > 0, let us consider a cut-off regular function ψ R (x) = ψ R (r) such that and let us set .
Hence, from the variational characterization of λ 1,p we deduce that as R → +∞, it is easy to see that given ǫ > 0 we can fix R > 0 such that For such a fixed value of R we can argue similarly as in the proof of (3.2) in Lemma 3.2 to obtain that, up to a subsequence in p, for p large enough. Hence the proof of Step 2 follows from (3.8), (3.9) and (3.10).
Corollary 3.4. Up to a subsequence ϕ 1,p → ϕ * 1 in L 2 (R 2 ) as p → +∞. Proof. By Lemma 3.2 and Theorem 3.3 we have that, passing to a subsequence as p → +∞, namely ϕ 1,p is a minimizing sequence for (3.1) and so the result follows from points ii) and iii) of Proposition 3.1.

Proof of Theorem 1.1
The proof of Theorem 1.1 follows the same strategy as in [1,7,4] and it is a consequence of Proposition 1.4, which is a particular case of Theorem 2.3 in [1]. Hence, to obtain Theorem 1.1, it is enough to prove the following: Proof. Since by an easy computation [4, pg. 14]), we can study the sign of which is the same as the sign of We show that, up to a subsequence, from which the conclusion follows.
In order to prove (4.1) we change the variable and, for any R > 0, we split the integral in the following way By Hölder inequality, the convergence of v + p to U in C 1 loc (R 2 ) up to a subsequence (see Lemma 2.2) and Corollary 3.4 we have, for R > 0 fixed: as p → +∞, up to a subsequence. For R sufficiently large the term {|x|>R} e U ϕ * 1 dx may be made arbitrary small since e U ∈ L 1 (R 2 ) and ϕ * 1 is bounded (Proposition 3.1-iv)).

A sufficient condition for (B)
Next property is a sufficient condition for (B): Lemma 5.1. Assume that there exists C > 0 such that for all p sufficiently large and all x ∈ Ω. Then condition (B) holds true up to a subsequence in p.
Proof. Let R > 0 fixed and let y ∈ Ω, |y − x + p | > Rµ + p , then for p large, by (5.1) and (B) follows, up to a subsequence in p, passing to the limit as R → +∞.
Condition (5.1) is a special case of a more general result that has been proved in [3] for any family (u p ) of solutions to (1. and we introduce the following properties: (P n 1 ) For any i, j ∈ {1, . . . , n}, i = j, is the solution of −∆U = e U in R 2 , U ≤ 0, U(0) = 0 and R 2 e U = 8π. Moreover for all p sufficiently large and all x ∈ Ω. Proof. By (A) Proposition 5.2 applies. Just observe that x 1,p = x + p and so when k = 1 property (P k 3 ) reduces to (5.1) and so the conclusion follows by Lemma 5.1.
In the following we use Theorem 5.3 to obtain condition (B) for suitable classes of solutions.
6. Proof of Theorem 1.2 Before proving Theorem 1.2 we observe that the existence of sign changing stationary solutions u p to (1.1) satisfying assumptions (1.3) and (1.4) has been proved for m ≥ 4 in [2] for p large. The proof uses the fact that the energy is decreasing along non constant solutions, and relies on constructing a suitable initial condition v 0 for problem (1.1) such that any stationary solution in the corresponding ω-limit set satisfies the energy estimate (1.4). This construction can be done for p large even without any symmetry assumption on Ω (see [2] for details). Anyway when Ω is a simply connected G-symmetric smooth bounded domain with |G| ≥ m also some qualitative properties of u p under condition (1.4) may be obtained (for instance the nodal line does not touch ∂Ω, it does not pass through the origin, etc, as shown in [2]). Then, in [3] a deeper asymptotic analysis of u p as p → +∞ has been done, showing concentration in the origin and a bubble tower behavior, when Ω is a simply connected G-symmetric smooth bounded domain with |G| ≥ me.
Here we do not require Ω to be simply connected.
Clearly assumption (1.4) is a special case of condition (A), hence in particular Proposition 5.2 holds. As before we assume w.l.o.g. that u p ∞ = u + p ∞ .
The proof of Theorem 1.2 follows then from the following As we will see Proposition 6.1 is a consequence of the general sufficient condition in Theorem 5.3. Hence in order to prove it we only need to show that k = 1, where the number k is the maximal number of families of points (x i,p ), i = 1, . . . , k, for which (P k 1 ), (P k 2 ) and (P k 3 ) hold, up to a subsequence, as in Proposition 5.2. When m = 4 the result has been already proved in [3, Proposition 3.6]. Here we show the general case (see also [3,Remark 4.6]). We start with the following: Let us fix i ∈ {1, . . . , k}. In order to simplify the notation we drop the dependence on i namely we set x p := x i,p and µ p := µ i,p .
Without loss of generality we can assume that either (x p ) p ⊂ N + p or (x p ) p ⊂ N − p . We prove the result in the case (x p ) p ⊂ N + p , the other case being similar.
Proof. The proof is the same as in [3, Proposition 3.6], we repeat it for completeness. Let us assume by contradiction that k > 1 and set x + p = x 1,p . For a family (x j,p ), j ∈ {2, . . . , k} by Lemma 6.2, there exists C > 0 such that |x 1,p | µ 1,p ≤ C and |x j,p | µ j,p ≤ C.

A special case in Theorem 1.2: the radial solutions
In this section we show that, when the domain Ω is the unit ball in R 2 , the unique (up to a sign) radial solution u p,K of (1.2) with K ≥ 2 nodal regions satisfies conditions (A) and (B).
Thus Theorem 1.1 applies to u p,K , namely we re-obtain the result already known from [4] through a different proof which does not rely on radial arguments.
Let us fix the number of nodal regions K ≥ 2. As before we assume w.l.o.g. that u p,K ∞ = u + p,K ∞ .
The main result is the following: Let Ω be the unit ball in R 2 and for K ≥ 2 let u p,K be the unique radial solution of (1.2) with K nodal domains. Then there exists m(= m(K)) > 0 for which the assumptions of Theorem 1.2 are satisfied.
Proof. In [4, Proposition 2.1] it has been proved that u p,K satisfies assumption (A) (by extending the arguments employed in [2] for the case with two nodal regions). Hence there exists C(= C(K)) such that p Ω |u p,K | p+1 dx ≤ C.