On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration

<jats:p>This paper deals with a class of Petrovsky system with nonlinear damping <jats:disp-formula><jats:alternatives><jats:tex-math> $$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{aligned}$$ </jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>w</mml:mi>
                    <mml:mrow>
                      <mml:mi>t</mml:mi>
                      <mml:mi>t</mml:mi>
                    </mml:mrow>
                  </mml:msub>
                  <mml:mo>+</mml:mo>
                  <mml:msubsup>
                    <mml:mi>Δ</mml:mi>
                    <mml:mi>B</mml:mi>
                    <mml:mn>2</mml:mn>
                  </mml:msubsup>
                  <mml:mi>w</mml:mi>
                  <mml:mo>−</mml:mo>
                  <mml:msub>
                    <mml:mi>k</mml:mi>
                    <mml:mn>2</mml:mn>
                  </mml:msub>
                  <mml:msub>
                    <mml:mi>Δ</mml:mi>
                    <mml:mi>B</mml:mi>
                  </mml:msub>
                  <mml:msub>
                    <mml:mi>w</mml:mi>
                    <mml:mi>t</mml:mi>
                  </mml:msub>
                  <mml:mo>+</mml:mo>
                  <mml:mi>a</mml:mi>
                  <mml:msub>
                    <mml:mi>w</mml:mi>
                    <mml:mi>t</mml:mi>
                  </mml:msub>
                  <mml:mo>|</mml:mo>
                  <mml:msub>
                    <mml:mi>w</mml:mi>
                    <mml:mi>t</mml:mi>
                  </mml:msub>
                  <mml:msup>
                    <mml:mo>|</mml:mo>
                    <mml:mrow>
                      <mml:mi>m</mml:mi>
                      <mml:mo>−</mml:mo>
                      <mml:mn>2</mml:mn>
                    </mml:mrow>
                  </mml:msup>
                  <mml:mo>=</mml:mo>
                  <mml:mi>b</mml:mi>
                  <mml:mi>w</mml:mi>
                  <mml:mo>|</mml:mo>
                  <mml:mi>w</mml:mi>
                  <mml:msup>
                    <mml:mo>|</mml:mo>
                    <mml:mrow>
                      <mml:mi>p</mml:mi>
                      <mml:mo>−</mml:mo>
                      <mml:mn>2</mml:mn>
                    </mml:mrow>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:disp-formula> on a manifold with conical singularity, where <jats:inline-formula><jats:alternatives><jats:tex-math>$\Delta _{\mathbb{B}}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>Δ</mml:mi>
                    <mml:mi>B</mml:mi>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula> is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary <jats:inline-formula><jats:alternatives><jats:tex-math>$x_{1}=0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>x</mml:mi>
                    <mml:mn>1</mml:mn>
                  </mml:msub>
                  <mml:mo>=</mml:mo>
                  <mml:mn>0</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula>. We first prove the global existence of solutions under conditions without relation between <jats:italic>m</jats:italic> and <jats:italic>p</jats:italic>, and establish an exponential decay rate. Furthermore, we obtain a finite time blow-up result for local solutions with low initial energy <jats:inline-formula><jats:alternatives><jats:tex-math>$E(0)< d$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>E</mml:mi>
                  <mml:mo>(</mml:mo>
                  <mml:mn>0</mml:mn>
                  <mml:mo>)</mml:mo>
                  <mml:mo><</mml:mo>
                  <mml:mi>d</mml:mi>
                </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>


Introduction
Due to the frequent occurrence of high order nonlinear wave equations in many branches of engineering, physics, chemistry, material science, and other sciences, the study of wave equations plays a key role in mathematical analysis. For more details, see [1,2]. In [3] and [4], the original Petrovsky model has the following form: where Ω ∈ R n is a bounded domain with a smooth boundary ∂Ω. Equation (1.1) is an important physical model that appears in many applications to mathematical physics as well as in the theory of vibrating plates, geophysics, and ocean acoustics [5,6]. Some further physical interpretations are given in [7,8].
For Equation (1.1), many results for global existence, nonexistence, and asymptotic behavior of solutions have been obtained [3][4][5][6][7][8][9][10][11]. Li et al. [3] studied problem (1.1)-(1.3) and derived that the solution is global without the relation between m and p. Moreover, the decay estimates of the energy function and the estimates of the lifespan of solution were given. Later, under suitable conditions decay estimates of the solutions for Equation (1.1) have been established by using Nakao's inequality in [4]. Messaoudi [9] proved the solution for problem (1.1)-(1.3) without w t blows up in finite time if p > m and the energy is negative. Wu [10] proved the blow-up result for problem (1.1)-(1.3) without w t if p > m and the energy is nonnegative. Recently, Chen et al. [11] proved that the solution of problem (1.1)-(1.3) without w t blows up with positive initial energy and claimed that the solution blows up in finite time for even vanishing initial energy for m = 2. More recently, Philippin et al. [12] used a differential inequality technique to obtain a lower bound on blow-up time for Equation (1.1) without w t . In recent years, lower bounds for the blowup time in a superlinear hyperbolic equation with damping term have been derived [13]. For other related works, we refer the readers to [14][15][16][17][18] and the references therein.
In 2011 to 2012, Chen et al. established the corresponding Sobolev inequality on the cone Sobolev spaces in [19,20]. And on this basis, they studied the initial boundary value problem of a semilinear parabolic equation on a manifold with conical singularity [21] and obtained the existence and nonexistence results by introducing a family of potential wells. Li et al. [22] proved the global existence, exponential decay, and finite time blow-up of solution for a class of semilinear pseudo-parabolic equations with conical degeneration. Recently, Alimohammady et al. [23] studied a class of semilinear degenerate hyperbolic equations on the cone Sobolev spaces w tt -B w + V (x)w + γ w t = g t (x)w|w| p-1 , x ∈ int B, t > 0, (1.4) w(t, x) = 0, x ∈ ∂B, t ≥ 0, (1.5) w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), x ∈ int B, (1.6) where B = [0, 1) × X, X is an (n -1)-dimensional closed compact manifold, which is regarded as the local model near the conical points and ∂B = {0} × X. B = (x 1 ∂ x 1 ) 2 + ∂ 2 x 2 + · · · + ∂ 2 x n . They discussed the invariance of some sets, global existence, nonexistence, and asymptotic behavior of solutions with initial energy J(w 0 ) < d by introducing a family of potential wells which was first proposed by Sattinger [24]. More works on equations with conical degeneration can be seen in the literature [25][26][27][28] and the references therein.
If we consider Equation (1.1) on a manifold with conical singularity, that is, when the standard Laplace operator of Equation (1.1) is replaced by Fuchsian-type Laplace operator B , what will happened for the initial boundary value problem? For this kind of Petrovsky equation with conical degeneration, the existence and nonexistence of global solutions to both the initial boundary value problem and the initial value problem remain open.
Inspired by the ideas of [3,4,23] and [29][30][31], we study the initial boundary value problem for the following Petrovsky equation: where w 0 (x), w 1 (x) are suitable initial data and k 2 , a, b, m, p are constants such that k 2 and b are positive, a is nonnegative, and m ≥ 2, 2 < p < 2n n-2 = p * , where p * is the critical Sobolev exponents. Here B is defined as above, and ν is the unit normal vector pointing toward the exterior of B. Moreover, the operator B in (1.7) is defined by (x 1 ∂ x 1 ) 2 +∂ 2 x 2 +· · ·+∂ 2 x n , which is an elliptic operator with conical degeneration on the boundary x 1 = 0 (we also called it Fuchsian-type Laplace operator), and the divergence operator div B is defined by In the neighborhood of ∂B we will use coordinates ( Our main aim in this paper is to find the existence and nonexistence of solutions for problem (1.7)-(1.9) with cone degeneration by introducing a family of potential wells. Firstly, under the condition of low initial energy, we establish the existence of global solution in the cone Sobolev space by a combination of Galerkin method and potential well theory. Then, using the energy perturbation technique, we obtain the exponential decay result of the global solution. Finally, we show that the solution of the problem blows up in a finite time and give the estimates for lower and upper bounds of blow-up time. It is worth mentioning that two types of lower bounds of the blow-up time T max for the weak solution of (1.7)-(1.9) are given, respectively.
The rest of this article is organized as follows. In Sect. 2, we recall the cone Sobolev spaces and the corresponding properties. In Sect. 3, we establish a global existence result and show the decay rates. In Sect. 4, we prove the blow-up properties of local solution.

Preliminaries
In this section, we recall the manifold with conical singularities and the cone Sobolev spaces which were introduced in [19,20] and introduce some lemmas and notations.
We assume that the manifold B has only one conical point on the boundary. Thus, near the conical point, we have a stretched manifold B associated with B. Here B = [0, 1) × X, ∂B = {0} × X and X is a closed compact manifold of dimension n -1. Also, in the neighborhood of the conical point, we use coordinates (x 1 , x ) = (x 1 , x 2 , . . . , x n ) for 0 ≤ x 1 < 1, x ∈ X.
with p, q ∈ (1, +∞) and 1 p + 1 q = 1, then we have the following Hölder inequality: In the sequel, for convenience we denote
From Lemma 2.2 and Lemma 2.3, we obtain the following lemma.

Global existence and energy decay
In this section, we discuss the global existence and decay of the solution for problem (1.7)-(1.9). Similar to the classical case, we introduce the following functionals on cone Sobolev spaceH 2, n 2 2,0 (B): We also define the energy function as follows: Finally, we introduce the potential well and the outside sets of the corresponding potential well

Remark 3.1 By (3.3) and Lemma 2.4, we know that
p λ p and C 0 is given in Lemma 2.4. A direct calculation shows that g(λ) has the maximum value at By the definition of g(λ) and J(w), we can give another definition of d as follows: (3.8) and the Nehari manifold Similar to the results in [29], one has 0 < d = inf w∈N J(w). The next lemma shows that our energy functional E(t) is a nonincreasing function along the solution of (1.7)-(1.9).

Lemma 3.1 E(t) is a nonincreasing function for t ≥ 0 and
Proof Multiplying (1.7) by w t and integrating it over B × [0, t), we obtain for t ≥ 0. Thus, the proof is completed.
It implies . This is a contradiction.
> λ 1 , as in case (i) we also deduce that Then w ∈ W for each t ≥ 0.
Proof When w = 0, we get w ∈ W easily, so we just need to prove the case w = 0. Since I(w 0 ) > 0, it follows from the continuity of w that for some interval near t = 0. Let T m > 0 be a maximal time (possibly T m = T) when (3.13) holds on [0, T m ). From (3.1)-(3.2), it follows that (3.14) By using (3.14), (3.3), and Lemma 3.1, we get Then, by Lemma 2.4 and (3.15), we obtain on t ∈ [0, T m ). Therefore, by using (3.2), we conclude that I(w) > 0 for all t ∈ [0, T m ). By repeating the procedure, T m is extended to T. The proof is completed.
, and E(0) < d, let w 0 ∈ W and w satisfy the assumption of Lemma 3.3. Then problem (1.7)-(1.9) admits a global weak solu- Proof Let {ω j (x)} be a system of base functions inH 2, n 2 2,0 (B). Now we construct the following approximate solution w s (x, t) of problem (1.7)-(1.9): Multiplying (3.18) by g js (t), summing for j (j = 1, 2, . . . , s), and integrating from 0 to t, we obtain dτ + E w s (t) < d, for sufficiently large s, which yields In (3.18), we fix j, letting s → ∞ and integrating from 0 to t. Then we have and ∀v ∈H  9). It is obvious that w(t) ∈ W for 0 ≤ t < ∞. Now, we use the following "modified" functional: .

Lemma 3.4
Let w satisfy the assumption of Theorem 3.1. For ε small enough, we have holds for two positive constants α 1 and α 2 .
Proof Making use of (3.23), straightforward computations lead to and in the same way, we get for ε small enough.

Theorem 3.2
Suppose that 2 ≤ m < m * = n-2 2n . Let w(x, t) satisfy the assumption of Theorem 3.1. Then we have the following decay estimates: where K and k are positive constants which will be defined later.
Proof From the definition of G(t), we get Using Lemma 2.2, we obtain Then, we will show that from the estimate of the last term in Choose ε so small that ε( p 2 + 1)c 2k 2 ≤ 0, εθ -1 ≤ 0. And choose suitable θ such that p -[ac(θ )C m 0 ( 2p p-2 E(0)) m-2 2 + ( p 2 -1)] > 0. Then from the above inequality, we obtain Then, by the relation between E(t) and G(t), we get We take ε small enough such that Integrating (3.43), we obtain By using (3.33) again, we get where K = α 2 G(0). This completes the proof.

Finite time blow-up of solution
In this section, we show that the solution of problem (1.7)-(1.9) blows up in finite time if p > m and E(0) < d. For this purpose, we first give the following lemma which will be used later.

(4.2)
Proof Let w 0 ∈ V , we have to prove that w(t) ∈ V for all t ∈ [0, T). We argue by contradiction. Assume that there exists t 0 ∈ [0, T) such that w(t 0 ) / ∈ V . This implies that .
By the continuity of w(t), there exists at least onet ∈ (0, t 0 ] such that .
In particular, the regularity of w(t) implies thatt ∈ (0, t 0 ]. Thus, we know and w(t) ∈ V for all t ∈ [0,t). We have two cases to consider. On the other hand, the fact that In this case, by recalling (3.8), we know that J(w(t)) ≥ d. Thus, E(t) ≥ d, which contradicts the fact that E(t) ≤ E(0) < d. Hence, in either case we conclude that w(t) ∈ V for all t ∈ [0, T). Since An elementary calculation shows , with 2 ≤ s ≤ p, (4.7) for any w ∈H . Therefore (4.7) follows. Now we introduce the following auxiliary function:  , with 2 ≤ s ≤ p, (4.9) for any w ∈H 2, n 2 2,0 (B).

9). Then a lower bound T for the lifespan t of w is given by
where 1 < α < 2 andc 2 ,c 3 are positive constants to be determined later.
Proof Now we want to derive a lower bound for the lifespan t of the blow-up solution.
To this end, we introduce the auxiliary function and compute a value T > 0 such that φ(t) remains bounded for t ∈ [0, T]. Clearly, T is a lower bound for t . Differentiating (4.39) and making use of the second Green's formula, we obtain in view of (1.7) Now we make use of Hölder's inequality to the first term on the right-hand side of (4. In the rest of the proof, we apply Young's inequality to the first term on the right-hand side of (4.42) with exponents α and α α-1 , where 1 < α < 2 is a constant. Thus we obtain Thus, we obtain the desired result.
In the following theorem, by means of a first order differential inequality technique, we obtain a lower bound for the blow-up time which is different from (4.38). Proof We introduce the auxiliary function and compute a valueT > 0 such that ψ(t) remains bounded for t ∈ [0,T]. ClearlyT is a lower bound for t . Differentiating (4.49) and making use of the second Green's formula, we obtain in view of (1.7) Making use of the Schwarz inequality leads to (4.51) Applying the Poincaré inequality, we obtain