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A necessary and sufficient condition of blow-up for a nonlinear equation
Boundary Value Problems volume 2023, Article number: 28 (2023)
Abstract
We investigate a nonlinear equation with quadratic nonlinearities, including a nonlinear model in Silva and Freire (J. Differ. Equ. 320:371–398, 2022). Using the classical energy estimate methods, we give a necessary and sufficient condition of blow-up of solutions to nonlinear equations. We answer a problem pointed out by Silva and Freire (J. Differ. Equ. 320:371–398, 2022).
1 Introduction
Silva and Freire [1] investigated in detail the following equation:
for which they considered continuation and persistence of solutions and necessary conditions for blow-up of a solution.
Equation (1.1) is related to the equation
where constants \(a,b,c\) satisfy \((ab,ac)\neq (0,0)\), and \(k\neq 0\) (see [2]). Under certain restrictions on the parameters \(a,b,c\), and k, the conserved currents, peakon solutions, and point symmetries are discussed in [2–4]. Obviously, when \(a=3, b=2,c=1\), and \(k=1\), Eq. (1.2) reduces to the standard Camassa–Holm equation [5]. If \(a=4, b=3,c=1\), and \(k=1\), then Eq. (1.2) becomes the Degasperis–Procesi model [6]. When \(a=4, b=3, c=1\), and \(k=2\), Eq. (1.2) reduces to the Novikov equation [7]. For \(a=b+c\), \(b\in \mathbb{R}\), \(c\neq 0\), and \(k>0\), if the initial value belongs to a suitable Besov space, the well-posedness of short-time solutions for Eq. (1.2) is investigated in [8]. Under certain restrictions on the constants \(a,b,c,k\), the global well-posedness for Eq. (1.2) is also established in Yan [8]. For real b, \(c=1\), and \(a=b+1\), the traveling wave solutions, the persistence properties, and unique continuation to Eq. (1.2) are considered by Guo et al. [9, 10] and Himonas and Thompson [11, 12]. Under different assumptions on the parameters \(a,b,c,k\) and the initial data, many useful dynamical properties for Eq. (1.2) can be found in [13–17].
We consider the following initial value problem:
where the constant \(m\in (-\infty , \infty )\). If \(m=1\), then the first equation in (1.3) becomes Eq. (1.1).
For problem (1.3) with \(m=1\), Silva and Freire [1] pointed out the following conjecture.
Conjecture
Let \(m=1, s>\frac{3}{2}, W_{0}(x)\in H^{s}(\mathbb{R})\), and lifespan \(T>0\). Then the solution \(W(t,x)\) of problem (1.3) blows up at finite time if and only if
The conjecture is presented on p. 396 in [1]. We will derive several estimates from problem (1.3) itself. Using the obtained estimates, we obtain two results: (1) If \(W_{0}(x)\in H^{s}(\mathbb{R}), s>\frac{3}{2}\), and the solution of problem (1.3) blows up, then \(\int _{0}^{T}|W_{x}(t,x)|\,dx=\infty \), where T is the lifespan of \(W(t,x)\) (2) If \(W_{0}(x)\in H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\), then \(\lim_{t\rightarrow T}\Vert W(t,\cdot )\Vert _{H^{s}}= \infty \) if and only if (1.4) holds. Our Theorem 3.2 demonstrates that the conjecture is right for any constant \(m\in (-\infty , \infty )\).
In Sect. 2, we present several lemmas, and in Sect. 3, we provide our main results and their proofs.
2 Several lemmas
Set \(\Lambda ^{2}=1-\partial _{x}^{2}\). Then \(\partial _{x}^{2}=1-\Lambda ^{2}\) and \(\Lambda ^{-2}=(1-\partial _{x}^{2})^{-1}\), and we have
Thus problem (1.3) becomes
Lemma 2.1
Let \(W_{0}\in H^{s}(\mathbb{R}) \) with \(s>\frac{3}{2}\). Then there is \(T=T(W_{0})>0\) such that problem (2.1) has a unique solution \(W(t,x)\), and
Using the Kato theorem [18], we can prove the well-posedness of local solutions for problem (2.1). In fact, the proof of well-posedness of a short-time solution for problem (2.1) is very similar to those of the famous Camassa–Holm and Degasperis–Procesi models (see [11, 15, 16]). Here we omit its proof.
Lemma 2.2
Suppose that \(s\geq 3\) and \(W(t,x)\in H^{s}(\mathbb{R})\). Then
Proof
SinceFootnote 1
we get (2.2). Similarly, we have
which leads to (2.3). □
Lemma 2.3
Let \(W_{0}(x)\in H^{s}(\mathbb{R})\) \((s>\frac{3}{2})\). Then
Proof
We only need to prove the first identity in (2.4). Since
and
by the Tonelli theorem we get
which finishes the proof. □
Lemma 2.4
([19])
If \(r\geq 0\) and \(f_{1},f_{2}\in H^{r}\cap L^{\infty}\), then
where the constant \(c>0\) depends only on r.
Lemma 2.5
([19])
Let \(f_{1}\in H^{r}\cap W^{1,\infty}\) \((r>0)\) and \(f_{2}\in H^{r-1}\cap L^{\infty}\). Then
where \([\Lambda ^{r},f_{1} ]=\Lambda ^{r}f_{1}-f_{1}\Lambda ^{r}\), and the constant \(c>0\) depends only on r.
Remark 1
Using the arguments in [8, 15], the lifespan T in Lemma 2.1 does not depend on the Sobolev index \(s>\frac{3}{2}\). Namely, for arbitrary \(s_{1}>s>\frac{3}{2}\) or \(s>s_{1}>\frac{3}{2}\), the maximal existence time for \(\Vert W\Vert _{H^{s}}\) and \(\Vert W\Vert _{H^{s_{1}}}\) is the same.
3 Main results
Theorem 3.1
Let \(W_{0}\in H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\), and suppose W satisfies problem (1.3) or problem (2.1). If the lifespan T of W is finite and
then
Proof
If \(s>\frac{3}{2}\), then using the operator \(\Lambda ^{s}W\Lambda ^{s}\), from problem (2.1) we obtain
which leads to
In fact, we have
from which we obtain
Employing the Cauchy–Schwarz inequality, (3.4), and Lemma 2.5, we acquire
which leads to
Similarly to the proof of (3.5), we have
Now Lemma 2.4 yields
Using inequalities (3.3), (3.5),(3.6), and (3.7) results in
where \(c>0\) is a constant. Using (3.8) yields
Suppose that \(\lim_{t\rightarrow T}\Vert W\Vert _{H^{s}}=\infty \). From (3.9) we have
which ends the proof. □
Theorem 3.2
Let \(W_{0}(x)\in H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\), and let T be the lifespan of solution \(W(t,x)\) for problem (2.1). If T is finite, then
if and only if
Proof
Let (3.10) hold. We will derive that (3.11) holds. Using Remark 1 and choosing \(s=3\), Lemma 2.1 ensures that there exists \(W(t,x)\in C([0,T), H^{3}(\mathbb{R}))\cap C^{1}([0,T), H^{2}( \mathbb{R}))\). We will employ the classical energy estimates. From problem (2.1) we acquire
Applying the first equation in (2.1) yields
Using Lemma 2.2 and (3.13), we have
Using (3.13) gives rise to
Applying integration by parts, (3.15), and Lemma 2.2, we have
Using (3.12), (3.14), and (3.16), we have
If (3.10) holds, then suppose that we can choose a positive constant M satisfying
Employing (3.17), (3.18), Lemma 2.3, \(\Lambda ^{-2}(W^{2})\geq 0\), and \(\Lambda ^{-2}(W_{x})^{2}\geq 0\), we have
Let
From (3.19) we obtain
which, together with the Gronwall inequality, yields
From (3.20) we obtain \(W(t,x)\in H^{2}(\mathbb{R})\), which, combined with Remark 1, is a contradiction to (3.10). Therefore we conclude that assumption (3.18) is not right.
Conversely, using \(\Vert W_{x}\Vert _{L^{\infty}}< c \Vert W\Vert _{H^{s}}\), if
then we derive that
The proof is completed. □
Availability of data and materials
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Notes
For any \(f\in L^{r}(\mathbb{R})\) with \(1\leq r\leq \infty \), we have \(\Lambda ^{-2}f(x)=\frac{1}{2}\int _{\mathbb{R}}e^{-|x-\eta |}f(\eta )\,d \eta \) (see Constantin and Escher [14]). If a function \(g\in H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\), then \(g(\pm \infty )=g'(\pm \infty )=g''(\pm \infty )=g^{[s]}(\pm \infty )=0\), where \([s]\) denotes the integer part of s (see [18]).
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This work is supported by the National Natural Science Foundation of China (No. 11471263).
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Liu, M., Tang, J. & Lai, S. A necessary and sufficient condition of blow-up for a nonlinear equation. Bound Value Probl 2023, 28 (2023). https://doi.org/10.1186/s13661-023-01716-3
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DOI: https://doi.org/10.1186/s13661-023-01716-3
MSC
- 35G25
- 35L05
Keywords
- Local strong solutions
- Nonlinear equation
- Blow-up
- Sufficient and necessary conditions