This article is concerned with the initial-boundary value problem for a equation of quasi-hyperbolic type with logarithmic nonlinearity. By applying the Galerkin method and logarithmic Sobolev inequality, we prove the existence of global weak solutions for this problem. In addition, by means of the concavity analysis, we discuss the nonexistence of global solutions in the unstable set and give the lifespan estimation of solutions.
Citation: Yaojun Ye, Qianqian Zhu. Existence and nonexistence of global solutions for logarithmic hyperbolic equation[J]. Electronic Research Archive, 2022, 30(3): 1035-1051. doi: 10.3934/era.2022054
[1] | Xiaoqiang Dai, Wenke Li . Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem. Electronic Research Archive, 2021, 29(6): 4087-4098. doi: 10.3934/era.2021073 |
[2] | Yi Cheng, Ying Chu . A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms. Electronic Research Archive, 2021, 29(6): 3867-3887. doi: 10.3934/era.2021066 |
[3] | Li-ming Xiao, Cao Luo, Jie Liu . Global existence of weak solutions to a class of higher-order nonlinear evolution equations. Electronic Research Archive, 2024, 32(9): 5357-5376. doi: 10.3934/era.2024248 |
[4] | Mohammad Kafini, Maher Noor . Delayed wave equation with logarithmic variable-exponent nonlinearity. Electronic Research Archive, 2023, 31(5): 2974-2993. doi: 10.3934/era.2023150 |
[5] | Xiao Su, Hongwei Zhang . On the global existence and blow-up for the double dispersion equation with exponential term. Electronic Research Archive, 2023, 31(1): 467-491. doi: 10.3934/era.2023023 |
[6] | Ling Xue, Min Zhang, Kun Zhao, Xiaoming Zheng . Controlled dynamics of a chemotaxis model with logarithmic sensitivity by physical boundary conditions. Electronic Research Archive, 2022, 30(12): 4530-4552. doi: 10.3934/era.2022230 |
[7] | Jorge A. Esquivel-Avila . Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28(1): 347-367. doi: 10.3934/era.2020020 |
[8] | Xu Zhao, Wenshu Zhou . Vanishing diffusion limit and boundary layers for a nonlinear hyperbolic system with damping and diffusion. Electronic Research Archive, 2023, 31(10): 6505-6524. doi: 10.3934/era.2023329 |
[9] | Vo Van Au, Jagdev Singh, Anh Tuan Nguyen . Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29(6): 3581-3607. doi: 10.3934/era.2021052 |
[10] | Gongwei Liu . The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28(1): 263-289. doi: 10.3934/era.2020016 |
This article is concerned with the initial-boundary value problem for a equation of quasi-hyperbolic type with logarithmic nonlinearity. By applying the Galerkin method and logarithmic Sobolev inequality, we prove the existence of global weak solutions for this problem. In addition, by means of the concavity analysis, we discuss the nonexistence of global solutions in the unstable set and give the lifespan estimation of solutions.
In this paper, we study the initial-boundary value problem for logarithmic hyperbolic equation of p−Laplacian type
utt+Δpu=|u|p−2uln|u|, (x,t)∈Ω×R+, | (1.1) |
u(x,0)=u0(x), ut(x,0)=u1(x), x∈Ω, | (1.2) |
u(x,t)=0, (x,t)∈∂Ω×R+, | (1.3) |
where
Δpu=−n∑i=1∂∂xi(|∂u∂xi|p−2∂u∂xi), p>2. | (1.4) |
Ω⊂RN is a bounded domain with smooth boundary ∂Ω, and parameter p fulfills
2<p<+∞, N≤p; 2<p≤NpN−p, N>p. | (1.5) |
The logarithmic nonlinearity arises in a lot of different areas of physics such as quantum mechanics and inflation cosmology, and is applied to nuclear physics, optics and geophysics [1,2,3,4,5,6,7,8]. Because of these special physical meanings, the research of evolution equations with logarithmic nonlinearity has attracted much attention.
When the logarithmic term in Eq (1.1) is replaced by nonlinear source term |u|r−2u, the Eq (1.1) becomes
utt+Δpu=|u|r−2u, (x,t)∈Ω×R+. | (1.6) |
In the case of p=2 and r>2, J. Ball [9] and M. Tsustsumi [10] obtained the blow-up solutions with negative initial energy. By using the concavity method, H. A. Levine and L. E. Payne [11] established the nonexistence of global weak solutions for Eq (1.6) with the conditions (1.2) and (1.3). Y. J. Ye [12] proved the global existence and blow-up result of solutions to the Eq (1.6) with initial-boundary value conditions. S. Ibrahim and A. Lyaghfouri [13] considered the Cauchy problem of (1.6), under appropriate assumptions, they established the finite time blow-up of solutions and, hence, extended a result by V. A. Galaktionov and S. I. Pohozaev [14]. For the Eq (1.6) with dissipative term, Y. J. Ye [15,16] studied the global solutions by constructing a stable set in W1,p0(Ω) and the decay property of solution by using an integral inequality [17].
S. A. Messaoudi and B. S. Said-Houari [18] considered the nonlinear hyperbolic type equation
utt+Δαu+Δβut−Δut+a|ut|m−2ut=b|u|p−2u, |
where a,b>0,α,β,m,p>2 and Ω is a bounded domain in RN(N≥1). Under suitable conditions on α,β,m,p, they proved a global nonexistence result of solutions with negative initial energy. For the nonlinear wave equation of p-Laplacian type
utt+Δpu−Δut+q(x,u)=f(x). |
C. Chen et al.[19] obtained the global existence and uniqueness of solutions and established the long-time behavior of solutions.
L. C. Nhan and T. X. Le [20] studied the existence and nonexistence of global weak solutions for a class of p-Laplacian evolution equations with logarithmic nonlinearity and gave sufficient conditions for the large time decay and blow-up of solutions. Later, Y. Z. Han et al. [21] also considered this problem. They studied global solutions and blow-up solution for arbitrarily high initial energy. For a mixed pseudo-parabolic p-Laplacian type equation with logarithmic term, under various assumptions about initial values, H. Ding and J. Zhou [22] proved the solution exists globally and blow up in finite time. Moreover, T. Boudjeriou [23] was concerned with the fractional p-Laplacian with logarithmic nonlinearity, by applying the potential well theory and a differential inequality, he proved the existence and decay estimates of global solutions and obtained the blow-up result of solutions.
T. Cazenave and A. Haraux [8] considered the following logarithmic wave equation
utt−Δu=uln|u|, | (1.7) |
they gave the existence of solutions for the Cauchy problem of Eq (1.7). P. Gorka [4] obtained the global existence of weak solutions for the initial-boundary value problem of Eq (1.7). K. Bartkowski and P. Gorka [7] proved the existence of classical solutions and weak solutions for the corresponding one dimensional Cauchy problem of Eq (1.7). In the case of 0<E(0)≤d, W. Lian et al. [24] proved the global existence of solution and obtained the blow-up of solution for the Eq (1.7) with the conditions (1.2) and (1.3).
In this paper, by means of the potential well theory and the concavity analysis method [25,26,27,28,29], we prove the global existence and blow-up of solutions of the problem (1.1)–(1.3).
For simplicity, we denote Lp(Ω) and L2(Ω) norm by ‖⋅‖p and by ‖⋅‖ respectively. The space W1,p0(Ω) norm ‖⋅‖W1,p0(Ω) is replaced by ‖∇⋅‖p.
At first, we define the weak solutions of the problem (1.1)–(1.3) and give a few known lemmas.
Definition 2.1. [30] If
u∈C([0,T),W1,p0(Ω)), ut∈C([0,T),L2(Ω)), utt∈C([0,T),W−1,p′(Ω)) |
and satisfies
∫Ωuttφdx+∫ΩΔpuφdx=∫Ω|u|p−2uln|u|φdx, |
then u(t) is called a weak solution of (1.1)–(1.3) on [0,T), where φ∈W1,p0(Ω).
Lemma 2.1. [31] Let q be a real number with 2≤q<+∞ if 2≤n≤p and 2≤q≤npn−p if 2<p<n. Then there exists a positive constant C depending on Ω,p and q such that ‖u‖q≤C‖∇u‖p.
Lemma 2.2. [30] Let B0,B,B1 be Banach spaces with B0⊆B⊆B1 and
X={u: u∈Lp([0,T];B0), ut∈Lq[0,T];B1)}, 1≤p,q≤+∞. |
Suppose that B0 is compactly embedded in B and that B is continuously embedded in B1, then (i) the embedding of X into Lp(0,T;B) is compact if p<+∞. (ii) the embedding of X into C([0,T];B) is compact if p=+∞ and q>1.
Lemma 2.3. [32] Assume that un(x) is a bounded sequence in Lq(Ω),1≤q<+∞, un(x)→u(x) a.e.. Then u(x)∈Lq(Ω) and un(x)→u(x) weakly converges in Lq(Ω).
Lemma 2.4. [33,34,35] (L2−logarithmic Sobolev inequality) If v∈H10(Ω), then
∫Ω|v|2ln|v|dx≤‖v‖2ln‖v‖+a22π‖∇v‖2−n2(1+lna)‖v‖2, ∀a>0. | (2.1) |
In order to deal with the logarithmic term |u|p−2uln|u| in Eq (1.1), we introduce the following Lp−logarithmic Sobolev inequality.
Lemma 2.5. [36] Let u∈W1,p0(Ω), then one has the inequality
∫Ω|u|pln|u|dx≤‖u‖ppln‖u‖p+(p−2)a24π‖u‖pp+a22π‖∇u‖pp−np(1+lna)‖u‖pp, | (2.2) |
where a>0 is a constant.
For convenience, in the following we are going to give the proof of Lemma 2.5.
Proof. By (2.1) in Lemma 2.4, we have
∫Ω|v|2ln|v|dx≤‖v‖2ln‖v‖+a22π‖∇v‖2−n2(1+lna)‖v‖2=∫Ω|v|2dx⋅ln(∫Ω|v|2dx)12+a22π∫Ω|∇v|2dx−n2(1+lna)∫Ω|v|2dx. | (2.3) |
Let v=up2 in (2.3), then we obtain
12∫Ω|u|pln|u|pdx≤∫Ω|u|pdx⋅ln(∫Ω|u|pdx)12+a22π∫Ω|∇up2|2dx−n2(1+lna)∫Ω|u|pdx=12‖u‖ppln‖u‖pp+a22π∫Ω|∇up2|2dx−n2(1+lna)‖u‖pp. | (2.4) |
By direct computation, we get
∇up2=p2up2−1⋅∇u=p2up−22⋅∇u. | (2.5) |
From (2.5) and Hölder inequality, we receive
∫Ω|∇up2|2dx=p24∫Ω|u|p−2|∇u|2dx≤p24(∫Ω|u|pdx)p−2p(∫Ω|∇u|pdx)2p=p24‖u‖p−2p‖∇u‖2p. | (2.6) |
By Young inequality XY≤Xαα+Yββ with α=pp−2, β=p2, we conclude that
p24‖u‖p−2p‖∇u‖2p≤p24(p−2p‖u‖pp+2p‖∇u‖pp)=p(p−2)4‖u‖pp+p2‖∇u‖pp. | (2.7) |
It follows from (2.4), (2.6) and (2.7) that
12∫Ω|u|pln|u|pdx≤12‖u‖ppln‖u‖pp+a22π(p(p−2)4‖u‖pp+p2‖∇u‖pp)−n2(1+lna)‖u‖pp, |
which implies
∫Ω|u|pln|u|dx≤‖u‖ppln‖u‖p+(p−2)a24π‖u‖pp+a22π‖∇u‖pp−np(1+lna)‖u‖pp. |
This completes the proof of Lemma 2.5.
Next, we define the following functionals
J(u)=1p‖∇u‖pp−1p∫Ω|u|pln|u|dx+1p2‖u‖pp | (2.8) |
and
K(u)=‖∇u‖pp−1p∫Ω|u|pln|u|dx, | (2.9) |
for u∈W1,p0(Ω). By (2.8) and (2.9), we have
J(u)=1p2‖u‖pp+1pK(u). | (2.10) |
We denote the energy functional by
E(t)=12‖ut‖2+1p‖∇u‖pp−1p∫Ω|u|pln|u|dx+1p2‖u‖pp=12‖ut‖2+J(u), | (2.11) |
for u∈W1,p0(Ω), t≥0.
E(0)=12‖u1‖2+1p‖∇u0‖pp−1p∫Ω|u0|pln|u0|dx+1p2‖u0‖pp=12‖u1‖2+J(u0) | (2.12) |
is the initial total energy.
Moreover, we define the Nehari manifold [37]
N={u∈W1,p0(Ω)/{0}: K(u)=0, ‖∇u‖p≠0}, |
the stable set
W={u∈W1,p0(Ω): K(u)>0, J(u)<d}∪{0} |
and the unstable set
U={u∈W1,p0(Ω): K(u)<0 J(u)<d}, |
where
d=inf{supθ≥0J(θu): u∈W1,p0(Ω), ‖∇u‖p≠0}. | (2.13) |
It is readily seen that the potential well depth d defined in (2.13) can also be characterized as
d=infu∈NJ(u). | (2.14) |
Lemma 2.6. Let u∈W1,p0(Ω) and ‖u‖p≠0, then we have
(a) limθ→0+J(θu)=0, limθ→+∞J(θu)=−∞; |
(b) K(θu)=θJ′(θu){>0, 0<θ<θ∗,=0, θ=θ∗,<0, θ∗<θ<+∞. | (2.15) |
Proof. (a) For u∈W1,p0(Ω),
J(θu)=θpp‖∇u‖pp−θpp∫Ω|u|pln|u|dx−θpp‖u‖pplnθ+θpp2‖u‖pp. |
It is easy to get from ‖u‖p≠0 that (a) is valid.
(b) An elementary calculation shows that
ddθJ(θu)=θp−1(‖∇u‖pp−∫Ω|u|pln|u|dx−‖u‖pplnθ). | (2.16) |
Let ddθJ(θu)=0, then we have
θ∗=exp(‖∇u‖pp−∫Ω|u|pln|u|dx‖u‖pp). | (2.17) |
It follows from (2.9) that
K(θu)=θp(‖∇u‖pp−∫Ω|u|pln|u|dx−‖u‖pplnθ). | (2.18) |
From (2.16), (2.17) and (2.18), the Eq (2.15) holds.
Lemma 2.7. Suppose that u∈W1,p0(Ω) and ‖∇u‖p≠0. Then d≥M, where M=1p2(2π)n2e2(n+p)−p22.
Proof. From Lemma 2.6 and Eq (2.10), one has
supθ≥0J(θu)=J(θ∗u)=1p2‖θ∗u‖pp+1pK(θ∗u)=1p2‖θ∗u‖pp. | (2.19) |
We get from Lemma 2.5 that
K(u)=‖∇u‖pp−1p∫Ω|u|pln|u|dx≥(1−a22π)‖∇u‖pp+(npln(ae)−(p−2)a24π−ln‖u‖p)‖u‖pp. |
Choosing a=√2π, we have
K(u)≥(npln(√2π e)−p−22−ln‖u‖p)‖u‖pp=[ln((2π)n2pe2(n+p)−p22p)−ln‖u‖p]‖u‖pp. | (2.20) |
It follows from K(θ∗u)=0 and (2.20) that
ln((2π)n2e2(n+p)−p22)−ln‖θ∗u‖pp≤0, |
which implies
‖θ∗u‖pp≥(2π)n2e2(n+p)−p22. | (2.21) |
Thus, we obtain from (2.19) and (2.21) that
supθ≥0J(θu)≥1p2(2π)n2e2(n+p)−p22. | (2.22) |
Thus, by (2.13) and (2.22), we conclude that d≥M>0.
In this section, we state and prove the global existence result for the problem (1.1)–(1.3).
Theorem 3.1. Assume that p satisfies (1.5). If u0∈W1,p0(Ω), u1∈L2(Ω) and 0<E(0)<M, K(u0)≥0, then there is a global weak solution u(x,t) to the problem (1.1)–(1.3) which meets u(x,t)∈L∞([0,+∞);W1,p0(Ω)), ut(x,t)∈L∞([0,+∞);L2(Ω)).
Proof. Assume that {ωj}∞j=1 is a basis of space W1,p0(Ω) and that Vk is the subspace of W1,p0(Ω) generated by {ω1,ω2,⋯,ωm}, m=1,2,⋯. We shall look for the approximate solutions um(t)=m∑j=1gjm(t)ωj with gjm(t)∈C2[0,T], ∀T>0. Here the functions gjm(t) fulfil the following system of equations
(umtt,ωj)+(Δpum,ωj)=(|um|p−2umln|um|,ωj), j=1,2,⋯,m | (3.1) |
with initial data
um(0)=u0m, umt(0)=u1m. | (3.2) |
Because W1,p0(Ω) is dense in L2(Ω), so there exist αjm and βjm such that
u0m=m∑j=1gjm(0)ωj=m∑j=1αjmωj→u0(x) strongly in W1,p0(Ω), m→∞, | (3.3) |
u1m=m∑j=1g′jm(0)ωj=m∑j=1βjmωj→u1(x) strongly in L2(Ω), m→∞. | (3.4) |
By Picard's iteration method, the solutions gjm(t) for the Cauchy problem (3.1)–(3.2) exist in t∈[0,tm), tm≤T. By the uniformly boundedness of functions gjm(t) and the extension theorem, these solutions gjm(t) exists in the whole interval [0,T].
Multiplying both sides of (3.1) by g′jm(t), summing on j from 1 to m and then integrating over [0,t], we obtain
Em(t)=12‖umt(t)‖2+J(um(t))=12‖umt(0)‖2+J(um(0))=Em(0)<M≤d. | (3.5) |
From (3.5), it is easy to verify
um(t)∈W, ∀t∈[0,T]. | (3.6) |
Assume that there exists a time t1∈(0,T) such that um(t1)∉W, then, by the continuity of um(t) on t, we get um(t1)∈∂W. Thus, we receive either
J(um(t1))=d, | (3.7) |
or
K(um(t1))=0, ‖∇um‖p≠0. | (3.8) |
From (3.5), we have J(um(t1))<d. Thus, the case (3.7) is impossible.
If (3.8) is valid, um(t1)∈N. From (2.13), we obtain J(um(t1))≥d. This contradicts with (3.5). Therefore, the case (3.8) is also impossible as well.
We deduce from (2.10), (3.5) and (3.6) that
M>J(um)=1p2‖um‖pp+1pK(um)>1p2‖um‖pp, | (3.9) |
which implies that
‖um‖pp<Mp2. | (3.10) |
Taking a=√π in (2.2), we have from Lemma 2.5, Eqs (2.8) and (2.9) that
‖∇um‖pp=2K(um)+2∫Ω|um|pln|um|dx−‖∇um‖pp=2pJ(um)−2p‖um‖pp−‖∇um‖pp+2∫Ω|um|pln|um|dx≤2pJ(um)−2p‖um‖pp+p−22‖um‖pp−npln(πe2)‖um‖pp+2‖um‖ppln‖um‖p≤2pJ(um)+p−22‖um‖pp+2‖um‖ppln‖um‖p<CM. | (3.11) |
Here CM=2pM+(p−2)p22M+2pMln(p2M). From (3.5), we have
‖umt‖2<2M. | (3.12) |
For u,v∈W1,p0(Ω), by (1.4), we have (Δpu,v)=∫Ω|∇u|p−2|∇u|⋅|∇v|dx). Hence, from Hölder inequality and (3.11), we obtain
‖Δpu‖W−1,p′(Ω)≤‖∇u‖p−1p<Cp−1pM. | (3.13) |
It follows from (3.10)–(3.13) that the following limitations are true.
um→u weakly star in L∞(0,T;W1,p0(Ω)), | (3.14) |
um→u weakly star in L∞(0,T;Lp(Ω)), | (3.15) |
umt→ut weakly star in L∞(0,T;L2(Ω)), | (3.16) |
Δpum→χ weakly star in L∞(0,T;W−1,p′(Ω)). | (3.17) |
Combining (3.15) and (3.16) with Lemma 2.2 yields
um→u strongly in C([0,T];L2(Ω)), | (3.18) |
which implies
|um|p−2umln|um|→|u|p−2uln|u| almost everywhere (x,t)∈Ω×(0,T). | (3.19) |
Let Ω1={x∈Ω: |um(x,t)≤1} and Ω2={x∈Ω: |um(x,t)≥1}, then by means of direct calculation, we know from Lemma 2.1 and Eq (3.11)
∫Ω||um|p−2umln|um|dx|p′=∫Ω1||um|p−2umln|um|dx|p′+∫Ω2||um|p−2umln|um|dx|p′≤[(p−1)e]−p′|Ω|+(n−pp(p−1))p′∫Ω2|um|npn−p≤[(p−1)e]−p′|Ω|+(n−pp(p−1))p′Cnpn−p‖∇um‖npn−pp≤LM, | (3.20) |
where 1p+1p′=1 and LM=[(p−1)e]−p′|Ω|+(n−pp(p−1))p′Cnpn−pCnn−pM. From Lemma 2.3, Eqs (3.19) and (3.20), we receive
|um|p−2umln|um|→|u|p−2uln|u| weakly in L∞(0,T;Lp′(Ω)). | (3.21) |
Now, we prove χ=Δpu. For this reason, multiplying both sides of (3.1) by an arbitrary smooth function φ(t)∈C2[0,T] and integrating over [0,T], we have
(umt(T),φ(T)ωj)+∫T0(Δpum,φ(t)ωj)dt=(umt(0),φ(0)ωj)+∫T0(umt,φ′(t)ωj)dt+∫T0(|um|p−2umln|um|,φ(t)ωj)dt. | (3.22) |
Taking the limitation of both sides of Eq (3.22) with j fixed and m→∞, we get
(ut(T),φ(T)ωj)+∫T0(χ,φ(t)ωj)dt=(ut(0),φ(0)ωj)+∫T0(ut,φ′(t)ωj)dt+∫T0(|u|p−2uln|u|,φ(t)ωj)dt. | (3.23) |
By (3.23), we have
(ut(T),ψ(T))+∫T0(χ,ψ(t))dt=(ut(0),ψ(0))+∫T0(ut,ψ′(t))dt+∫T0(|u|p−2uln|u|,ψ(t))dt, | (3.24) |
for every ψ∈L2(0,T;W1,p0(Ω)), ψ′∈L2(0,T;L2(Ω)). In particular, Setting ψ=u in (3.24), we obtain
(ut(T),u(T))+∫T0(χ,u)dt=(ut(0),u(0))+∫T0‖ut(t)‖2dt+∫T0(|u|p−2uln|u|,u)dt. | (3.25) |
On the other hand, multiplying both sides of (3.1) by gjm(t), summing on j from 1 to m and integrating over [0,T], we get
(umt(T),um(T))+∫T0(Δpum,um)dt=(umt(0),um(0)+∫T0‖umt‖2dt+∫T0(|um|p−2umln|um|,um)dt. | (3.26) |
Taking the inferior limitation on both sides of (3.26) as m→∞, we have
(ut(T),u(T))+limm→∞inf∫T0(Δpum,um)dt≤(ut(0),u(0))+∫T0‖ut‖2dt+∫T0(|u|p−2uln|u|,u)dt. | (3.27) |
We conclude from (3.25) and (3.27) that
limm→∞inf∫T0(Δpum,um)dt≤∫T0(χ,u)dt. | (3.28) |
By the monotonicity of operator Δp, we have
∫T0(Δpum−Δpv,um−v)dt≥0, ∀v∈L∞(0,T;W1,p0(Ω)). | (3.29) |
We get from (3.28) and (3.29) that
limm→∞inf∫T0(Δpum−Δpv,um−v)dt≤∫T0(χ,u−v)dt−∫T0(Δpv,u−v)dt=∫T0(χ−Δpv,u−v)dt. | (3.30) |
Combining (3.29) with (3.30) yields that
∫T0(χ−Δpv,u−v)dt≥0. | (3.31) |
Let v=u−λω, then, by (3.31), we obtain
λ∫T0(χ−Δp(u−λω),ω)dt≥0, | (3.32) |
for any ω∈Lp(0,T;W1,p0(Ω)) and any real number λ.
As λ>0,λ→0, from (3.32) and the hemicontinuity of operator Δp, we conclude that
∫T0(χ−Δpu,ω)dt≥0. | (3.33) |
Similarly, when λ<0,λ→0, we have
∫T0(χ−Δpu,ω)dt≤0. | (3.34) |
Thus, for all ω∈Lp(0,T;W1,p0(Ω)), we deduce from (3.33) and (3.34) that
∫T0(χ−Δpu,ω)dt=0, | (3.35) |
which implies that χ=Δpu.
Next, we prove above solution u(x,t) satisfies (1.2), i.e., u(x,0)=u0(x), ut(x,0)=u1(x).
We conclude from Eqs (3.15), (3.16) and Lemma 1.2 that u(t):[0,T]→L2(Ω) is continuous. Therefore, um(0)→u(0) weakly in L2(Ω). According to (3.3), one has u(0)=u0.
To prove ut(0)=u1, let ξ(t) be a smooth function with ξ(0)=1,ξ(T)=0. Noting
∫T0(umtt,ξωj)dt=−∫T0(umt,ξtωj)dt−(umt(0),ξ(0)ωj). |
For given j, as m→∞, we get in the distribution sense
∫T0(utt,ξωj)dt=−∫T0(ut,ξtωj)dt−(ut(0),ξ(0)ωj) | (3.36) |
in D′([0,T]). On the other hand,
∫T0(umtt,ξωj)dt=∫T0[(−Δpum,ξωj)+(|um|p−2umln|um|,ξωj)]dt |
converges to
∫T0[(−Δpu,ξωj)+(|u|p−2uln|u|,ξωj)]dt=∫T0(utt,ξωj)dt |
as m→∞. Therefore,
∫T0(utt,ξωj)dt=−∫T0(ut,ξtωj)dt−(u1,ξ(0)ωj). | (3.37) |
From (3.36) and (3.37), we have (ut(0),ωj)=(u1,ωj). By the density of{ωj}mj=1 in L2(Ω), we get ut(0)=u1. This completes the proof of Theorem 3.1.
For the case of K(u0)≥0 and E(0)=M≤d, the global existence result of solutions to the problem (1.1)–(1.3) reads as follows:
Theorem 3.2. Assume that p fulfils (1.5). If u0∈W1,p0(Ω), u1∈L2(Ω) and E(0)=M≤d, K(u0)≥0, then there exists a global weak solution u(x,t) for the problem (1.1)–(1.3) which satisfies u(x,t)∈L∞([0,+∞);W1,p0(Ω)), ut(x,t)∈L∞([0,+∞);L2(Ω)).
Proof. For the case ‖∇u0‖p≠0, let us suppose that ρk=1−1k and u0k=ρku0, k≥2. The problem (1.1)–(1.3) can be written as follows:
{utt+Δpu=|u|p−2uln|u|, (x,t)∈Ω×R+,u(x,0)=u0k(x), ut(x,0)=u1(x), x∈Ω,u(x,t)=0, (x,t)∈∂Ω×R+. | (3.38) |
From K(u0)≥0 and Lemma 2.6, we have θ∗=θ∗(u0)≥1. Accordingly, we get K(u0k)>0. By (2.3), we obtain
0<J(u0k)=1p2‖u0k‖pp+1pK(u0k)<J(u0). | (3.39) |
Therefore, we receive
0<Ek(0)=12‖u1‖2+J(u0k)<12‖u1‖2+J(u0)=E(0)=M≤d, |
which implies that u0k∈W.
For each k, by Theorem 3.1, there exists a global weak solution uk(t) of the problem (3.38) such that uk(t)∈L∞([0,+∞);W1,p0(Ω)), ukt(t)∈L∞([0,+∞);L2(Ω)) and
(ukt,v)+∫t0(Δpuk,v)ds=(u1,v)+∫t0(|uk|p−2ukln|uk|,v)ds | (3.40) |
for any v∈W1,p0(Ω).
In addition,
Ek(t)=12‖ukt‖2+J(uk)=12‖u1‖2+J(u0k)=Ek(0)<M≤d. | (3.41) |
By using (3.41) and combining with the same argument as (3.6), we can prove uk(t)∈W.
For the case ‖∇u0‖p=0, we get J(u0)=0 by K(u0)≥0. Thus, we have E(0)=12‖u1‖2+J(u0)=12‖u1‖2=M. Let ρk=1−1k, u1k=ρku1(x), k≥2, we consider the following problem
{utt+Δpu=|u|p−2uln|u|, (x,t)∈Ω×R+,u(x,0)=u0(x), ut(x,0)=u1k(x), x∈Ω,u(x,t)=0, (x,t)∈∂Ω×R+. | (3.42) |
Noting
0<Ek(0)=12‖u1k‖2+J(u0)=12‖ρku1‖2<12‖u1‖2=M. | (3.43) |
By Eq (3.43) and Theorem 3.1, there is a global weak solution uk(t) for the problem (3.42) such that uk(t)∈L∞(0,+∞;W1,p0(Ω)), ukt(t)∈L∞(0,+∞;L2(Ω)) and uk(t)∈W for each k.
The remainder of the proof for Theorem 3.2 is the same as those of Theorem 3.1. Here, we omit them.
Lemma 4.1. [38,39] If nonnegative function Φ(t)∈C2 satisfies
Φ(t)Φ″(t)−(1+ρ)Φ′(t)2≥0, |
for Φ(0)>0, Φ′(0)>0 and ρ>0, then there exists a time T∗ such that 0<T∗≤Φ(0)ρΦ′(0) and limt→T−∗Φ(t)=+∞.
Lemma 4.2. Suppose that u(t) is a solution of (1.1)–(1.3). If u0∈U and E(0)<d, then u(t)∈U and E(t)<d, ∀t≥0.
Proof. From the conservation of energy, we obtain E(t)=E(0)<d. From (2.11), we get
J(u)≤E(t)<d. | (4.1) |
Assume that there is t∗∈[0,+∞) such that u(t∗)∉U, then by continuity of K(u(t)) on t, we obtain K(u(t∗))=0. That means u(t∗)∈N. From (2.14), we have J(u(t∗))≥d, which is contradiction with (4.1). Therefore, the conclusion in Lemma 4.2 holds.
Theorem 4.1. Suppose that 0<E(0)<d and ∫Ωu0u1dx>0, then there is no global weak solution u(t) to the problem (1.1)–(1.3). Namely, there exists a time T∗ such that limt→T−∗‖u(t)‖2=+∞, where the lifespan T∗ is estimated by 0<T∗<4Ψ(0)(p−2)Ψ′(0), Ψ(t) is given in (4.19).
Proof. By u0∈U, E(0)<d and Lemma 4.2, we get u∈U. Thus,
K(u)=‖∇u‖pp−∫Ω|u|pln|u|dx<0. | (4.2) |
From (2.13) and (2.19), we have
d≤supθ≥0J(θu)=1p2‖θ∗u‖pp. | (4.3) |
We deduce from (2.17), (4.2) and (4.3) that
d≤1p2‖u‖pp. | (4.4) |
Let
Ψ(t)=‖u(t)‖2=∫Ωu2dx. | (4.5) |
Then there is a real number α>0, which satisfies
Ψ(t)≥α>0. | (4.6) |
By differentiating on both sides of (4.5), we get
Ψ′(t)=2∫Ωuutdx. | (4.7) |
From (4.7), we obtain
Ψ″(t)=2‖ut‖2+2∫Ωuuttdx. | (4.8) |
Combining (1.1) with (4.8), we get
Ψ″(t)=2(‖ut(t)‖2+∫Ω|u|pln|u|dx−‖∇u‖pp)=2[‖ut(t)‖2−K(u)]. | (4.9) |
By u∈U and (4.9), we receive Ψ″(t)>0. Combining (4.5), (4.7) and (4.9), we get
Ψ(t)Ψ″(t)−p+24Ψ′(t)2=2Ψ(t)[‖ut(t)‖2+∫Ω|u|pln|u|dx−‖∇u‖pp]−(p+2)Ψ(t)‖ut(t)‖2+(p+2)Υ(t), | (4.10) |
where
Υ(t)=‖u(t)‖2⋅‖ut(t)‖2−(∫Ωuutdx)2. | (4.11) |
By Cauchy-Schwarz inequality, we get
(∫Ωuutdx)2≤‖u(t)‖2‖ut(t)‖2. | (4.12) |
This inequality (4.12) guarantees Υ(t)≥0. By (4.10), we have
Ψ(t)Ψ″(t)−p+24Ψ′(t)2≥Ψ(t)Π(t), | (4.13) |
where
Π(t)=−p‖ut‖2+2∫Ω|u|pln|u|dx−2‖∇u‖pp. | (4.14) |
From (2.11) and (4.14), we obtain
Π(t)=−2pE(t)+2p‖u‖pp. | (4.15) |
By (4.4), (4.15) and E(t)=E(0)<d, we get
Π(t)≥−2pE(0)+2pd=2p[d−E(0)]>0. | (4.16) |
Therefore, there exists β>0 such that
Π(t)≥β>0. | (4.17) |
Combining (4.6), (4.13) and (4.17), we conclude that
Ψ(t)Ψ″(t)−p+24Ψ′(t)2≥αβ>0, ∀t≥0. | (4.18) |
Let ρ=p−24>0, then, by the differential inequality (4.18) and Lemma 4.1, one has
0<T∗<4Ψ(0)(p−2)Ψ′(0), | (4.19) |
and
limt→T−∗Ψ(t)=+∞. | (4.20) |
From (4.5) and (4.20), we have limt→T−∗‖u(t)‖2=+∞.
This completes the proof of Theorem 4.1.
By applying Galerkin method and Lp-Sobolev logarithmic inequality, and combining with the potential well theory, we prove the global existence result of solutions in this paper. Namely, assume that p satisfies (1.5). If u0∈W1,p0(Ω), u1∈L2(Ω) and 0<E(0)≤M, K(u0)≥0, then there is a global weak solution u(x,t) of the problem (1.1)–(1.3). Meanwhile, under the condition of positive initial energy, by using the concavity analysis method, we establish the finite time blow-up result of solutions and give the lifespan estimate of solutions. The result read as follows: If 0<E(0)<d and ∫Ωu0u1dx>0, then the solutions of the problem (1.1)–(1.4) blows up in finite time and the lifespan T∗ is estimated by 0<T∗<4Ψ(0)(p−2)Ψ′(0).
The authors would like to thank the reviewers and editors for their help to improve the quality of this article. Moreover, this research was supported by Natural Science Foundation of Zhejiang Province (No. LY17A010009).
The authors declare that there is no conflict of interests regarding the publication of this paper.
[1] |
H. Buljan, A. Siber, M. Soljacic, T. Schwartz, M. Segev, D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E, 68 (2003), 036607. https://doi.org/10.1103/PhysRevE.68.036607 doi: 10.1103/PhysRevE.68.036607
![]() |
[2] | S. De Martino, M. Falanga, C. Godano, G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys. Lett., 63 (2003), 472–475. |
[3] | W. Krolikowski, D. Edmundson, O. Bang, Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E, 61 (2000), 3122–3126. https://doi.org/10.1103/PhysRevE.61.3122 |
[4] | P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Pol. B, 40 (2009), 59–66. |
[5] | I. Bialynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Pol. Sci. Ser. Sci. Phys. Astron., 23 (1975), 461–466. |
[6] | I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62–93. https://doi.org/10.1016/0003-4916(76)90057-9 |
[7] |
K. Bartkowski, P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A, 41 (2008), 355201, 11 pp. https://doi.org/10.1088/1751-8113/41/35/355201 doi: 10.1088/1751-8113/41/35/355201
![]() |
[8] | T. Cazenave, A. Haraux, Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21–51. |
[9] |
J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Q. J. Math., 28 (1977), 473–486. https://doi.org/10.1093/qmath/28.4.473 doi: 10.1093/qmath/28.4.473
![]() |
[10] | M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173–193. |
[11] |
H. A. Levine, L. E. Payne, Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl., 55 (1976), 329–334. https://doi.org/10.1016/0022-247X(76)90163-3 doi: 10.1016/0022-247X(76)90163-3
![]() |
[12] |
Y. J. Ye, Existence and nonexistence of solutions of the initial-boundary value problem for some degenerate hyperbolic equation, Acta Math. Sci., 25B (2005), 703–709. https://doi.org/10.1016/S0252-9602(17)30210-2 doi: 10.1016/S0252-9602(17)30210-2
![]() |
[13] |
S. Ibrahim, A. Lyaghfouri, Blow-up solutions of quasilinear hyperbolic equations with critical Sobolev exponent, Math. Modell. Nat. Phenom., 7 (2012), 66–76. https://doi.org/10.1051/mmnp/20127206 doi: 10.1051/mmnp/20127206
![]() |
[14] |
V. A. Galaktionov, S. I. Pohozaev, Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Anal. TMA, 53 (2003), 453–466. https://doi.org/10.1016/S0362-546X(02)00311-5 doi: 10.1016/S0362-546X(02)00311-5
![]() |
[15] | Y. Ye, Exponential decay of energy for some nonlinear hyperbolic equations with strong dissipation, Adv. Differ. Equations, (2010), 1–12. https://doi.org/10.1186/1687-1847-2010-357404 https://doi.org/10.1155/2010/357404 |
[16] |
Y. J. Ye, Global existence and asymptotic behavior of solutions for some nonlinear hyperbolic equation, J. Inequal. Appl., 2010 (2010), 1–10. https://doi.org/10.1155/2010/895121 doi: 10.1155/2010/895121
![]() |
[17] | V. Komornik, Exact Controllability and Stabilization: the Multiplier Method, Paris, 1994. |
[18] |
S. A. Messaoudi, B. S. Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms, Math. Methods Appl. Sci., 27 (2004), 1687–1696. https://doi.org/10.1002/mma.522 doi: 10.1002/mma.522
![]() |
[19] | C. Chen, H. Yao, L. Shao, Global existence, uniqueness, and asymptotic behavior of solution for p-Laplacian type wave equation, J. Inequal. Appl., 2010 (2010), 1–13. |
[20] |
L. C. Nhan, T. X. Le, Global solution and blow-up for a class of p-Laplacian evolution equations with logarithmic nonlinearity, Acta Appl. Math., 151 (2017), 149–169. https://doi.org/10.1016/j.camwa.2017.02.030 doi: 10.1016/j.camwa.2017.02.030
![]() |
[21] |
Y. Z. Han, C. L. Cao, P. Sun, A p-Laplace equation with logarithmic nonlinearity at high initial energy level, Acta Appl. Math., 164 (2019), 155–164. https://doi.org/10.1007/s10440-018-00230-4 doi: 10.1007/s10440-018-00230-4
![]() |
[22] |
H. Ding, J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), 393–420. https://doi.org/10.1080/00036811.2019.1695784 https://doi.org/10.1080/00036811.2019.1695784 doi: 10.1080/00036811.2019.1695784
![]() |
[23] |
T. Boudjeriou, Global existence and blow-Up for the fractional p-Laplacian with logarithmic nonlinearity, Mediterr. J. Math., 162 (2020), 1–24. https://doi.org/10.1007/s00009-020-01584-6 doi: 10.1007/s00009-020-01584-6
![]() |
[24] |
W. Lian, M. S. Ahmed, R. Z. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239–257. https://doi.org/10.1016/j.na.2019.02.015 doi: 10.1016/j.na.2019.02.015
![]() |
[25] | Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differ. Equations, 192 (2003), 155–169. |
[26] |
L. E. Payne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273–303. https://doi.org/10.1007/BF02761595 doi: 10.1007/BF02761595
![]() |
[27] | D. H. Sattinger, On global solutions for nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148–172. https://doi.org/10.1007/BF00250942 |
[28] | Y. C. Liu, J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal. TMA, 64 (2006) 2665–2687. https://doi.org/10.1016/j.na.2005.09.011 |
[29] |
L. Wang, H. Garg, Algorithm for multiple attribute decision-making with interactive archimedean norm operations under pythagorean fuzzy uncertainty, Int. J. Comput. Intell. Syst., 14 (2021), 503–527. https://doi.org/10.2991/ijcis.d.201215.002 doi: 10.2991/ijcis.d.201215.002
![]() |
[30] | J. L. Lions, Quelques Mthodes de Rsolution des Problmes aux Limites Nonlinaires, Paris, 1969. |
[31] | O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uralyseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, (1967), 23. |
[32] | S. M. Zheng, Nonlinear Evolution Equations, Chapman and Hall/CRC, 2004. |
[33] |
H. Chen, P. Luo, G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84–98. https://doi.org/10.1016/j.jmaa.2014.08.030 doi: 10.1016/j.jmaa.2014.08.030
![]() |
[34] |
H. Chen, S. Y. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equations, 258 (2015), 4424–4442. https://doi.org/10.1016/j.jde.2015.01.038 doi: 10.1016/j.jde.2015.01.038
![]() |
[35] |
L. Gross, Logarithmic Sobolev inequalities, Am. J. Math., 97 (1975), 1061–1083. https://doi.org/10.2307/2373688 doi: 10.2307/2373688
![]() |
[36] |
M. D. Pino, J. Dolbeault, I. Gentil, Nonlinear diffusions, hypercontractivity and the optimal Lp− Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl., 293 (2004), 375–388. https://doi.org/10.1016/j.jmaa.2003.10.009 doi: 10.1016/j.jmaa.2003.10.009
![]() |
[37] |
F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincaré-AN, 23 (2006), 185–207. https://doi.org/10.1016/j.anihpc.2005.02.007 doi: 10.1016/j.anihpc.2005.02.007
![]() |
[38] | H. A. Levine, Some nonexistence and instability theorems for formally parabolic equations of the form Put=−Au+F(u), Arch. Ration. Mech. Anal., 51 (1973), 371–386. |
[39] | V. K. Kalantarov, O. A. Ladyzhenskaya, The occurrence of collapse for quasi-linear equation of parabolic and hyperbolic typers, J. Sov. Math., 10 (1978), 53–70. |
1. | Amir Peyravi, Lifespan estimates and asymptotic stability for a class of fourth-order damped p-Laplacian wave equations with logarithmic nonlinearity, 2023, 29, 1405-213X, 10.1007/s40590-023-00570-8 | |
2. | Nazlı Irkıl, On the p-Laplacian type equation with logarithmic nonlinearity: existence, decay and blow up, 2023, 37, 0354-5180, 5485, 10.2298/FIL2316485I | |
3. | Bingchen Liu, Mengyao Liu, Critical blow-up exponent for a doubly dispersive quasilinear wave equation, 2024, 75, 0044-2275, 10.1007/s00033-024-02296-7 |