Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation

Definition 1

(Weak solution). Function u = u(x, t) is called a weak solution of problem (1.1) on Ω × [0, T), if u ∈ L^∞(0, T; H¹(Ω)) with u_t ∈ L²(0, T; H¹(Ω)) satisfies

1. ∫0t∫Ωuτφ+∇u∇φ+∇uτ∇φ−|u|p−1u−∫Ω−|u|p−1udxφdxdτ=0 (2.7)

   for any φ ∈ L^∞(0, T;H¹(Ω));

2. u(x, 0) = u₀(x) in H¹(Ω);

3. for 0 ≤ t < T,

   ∫0t∥uτ∥H12dτ+J(u)=J(u0). (2.8)

   Here, we show that the nonlocal term can give us a conservation law.

Lemma 2

(Conservation Law). Let u be the weak solution of problem (1.1). If ∫_Ω u₀(x)dx = 0, then the integral of u is conserved, that is ∫_Ω u(x, t)dx = 0.

Proof

For the weak solution defined by (2.7), we can take φ = 1 as the test function, then by noticing

we can obtain ∫_Ω u(x, t)dx = ∫_Ω u₀(x)dx = 0.□

Definition 3

(Maximal existence time). Let u(t) be a weak solution of problem (1.1). We define the maximal existence time T of u(t) as follows

1. If u(t) exists for 0 ≤ t < ∞, then T = +∞.

2. If there exists a t₀ ∈ (0, ∞) such that u(t) exists for 0 ≤ t < t₀, but doesn’t exist at t = t₀, then T = t₀.

Here, we give a estimate of nonlinear term ∣u∣^p−1u by Gâteaux derivative.

Lemma 4

If p satisfies (H), for any u₁(x, t), u₂(x, t) with (x, t) ∈ Ω × [0, T], ∣u₁∣ + ∣u₂∣ > 0 and u₁ ≠ u₂, there holds

Proof

Let F(u): = ∣u∣^p−1u. Define ψ(x, t): = u₁ − u₂ for (x, t) ∈ Ω × [0, T]. By the property of Gâteaux derivative, we know

for s ∈ (0, 1), which together with the definition of Gâteaux derivative

gives

□

For the Nehari functional I(u), we have the following properties.

Lemma 5

(Relations between I(u) and ∥∇ u∥₂). Assume u ∈ H¹(Ω).

1. If 0 < ∥∇ u∥₂ < r₀, then I(u) > 0;

2. If I(u) < 0, then ∥∇ u∥₂ > r₀;

3. If I(u) = 0 and ∥∇ u∥₂ ≠ 0, then ∥∇ u∥₂ ≥ r₀,

where

C_* is the best Sobolev constant for H¹(Ω) ↪ L^p+1(Ω) and

Proof

1. From 0 < ∥∇ u∥₂ < r₀, we have

   ∥u∥p+1p+1≤C∗p+1∥∇u∥2p+1=C∗p+1∥∇u∥2p−1∥∇u∥22<∥∇u∥22,

   which implies I(u) > 0.

2. From I(u) < 0 it implies that ∥∇ u∥₂ ≠ 0. Thus

   ∥∇u∥22<∥u∥p+1p+1≤C∗p+1∥∇u∥2p+1=C∗p+1∥∇u∥2p−1∥∇u∥22

   gives ∥∇ u∥₂ > r₀.

3. If I(u) = 0, ∥∇ u∥₂ ≠ 0, then from

   ∥∇u∥22=∥u∥p+1p+1≤C∗p+1∥∇u∥2p+2=C∗p+1∥∇u∥2p−1∥∇u∥22,

   we get ∥∇ u∥₂ ≥ r₀.

Lemma 6

(Properties of J(λu), [31]). Let u ∈ H¹(Ω) and ∥∇u∥22≠0. Then

1. limλ→0J(λu)=0,limλ→+∞J(λu)=−∞;

2. for λ ∈ (0, ∞), there exists a unique λ^* = λ^*(u), such that

   ddλJ(λu)∣λ=λ∗=0;

3. the point λ = λ^* is the maximum point of J(λu), J(λu) is increasing on λ ∈ [0, λ^*] and decreasing on λ ∈ [λ^*, ∞);

4. I(λ^* u) = 0, I(λu) > 0 for 0 < λ < λ^* and I(λu) < 0 for λ^* < λ < ∞.

Lemma 7

(Depth d of potential well). Suppose that (H) holds. Then the potential well depth

Proof

By the definition of J(u), I(u) and the fact that I(u) = 0, we have

which implies J(u) > 0.

By Lemma 5 (iii), there holds

which says

□

Remark 1

For condition J(u₀) < d, a simple computation induces that

Theorem 8

(Local solution). Suppose that (H) and u₀ ∈ H¹(Ω). Then there exist T > 0 and a unique solution of (1.1) over [0, T]. Moreover, if

then

Proof

The proof of this theorem is divided into three steps. The first step is to show that the existence and uniqueness of solution to the linear problem corresponding to problem (1.1) by Galerkin method. The second step is to prove the existence and uniqueness of the local solution to problem (1.1) based on first step by the Contraction Mapping Principle. And the third step is to get the assertion (3.11).

1. Take u₀ ∈ H¹(Ω) and let R2=2∥∇u0∥22. For every T > 0 consider the space 𝓗 = C([0, T];H¹(Ω)) endowed with the norm

   ∥u∥H2=maxt∈[0,T]∥∇u∥22 (3.12)

   and the set

   HT:=u∈H|∥u∥H≤R.

   Next for every T > 0 and u ∈ 𝓗_T, we shall respectively prove the existence and uniqueness of solution to the following linear problem by Galerkin method

   vt−Δv−Δvt=|u|p−1u−∫Ω−|u|p−1udx,inΩ×(0,T),v(x,0)=u0(x)≢0,inΩ,∫Ω−v0dx=0,inΩ,∂v∂ν(x,t)=0,on∂Ω×(0,T). (3.13)

2. Fix a positive integer m, assume that {ω_j(x)} is an orthogonal complete basis in H¹(Ω) and L²(Ω) such that 𝓦_m = Span{ω₁, ⋯, ω_m} and ∥ω_j∥₂ = 1 for all j. Denote by λ_j the related eigenvalues of ω_j(x). Let

   um0(x)=∑j=1m∫Ω∇u0∇ωjdxωj

   such that u_m0 ∈ 𝓦_m and u_m0 → u₀ in H¹(Ω) as m → ∞, then u_m0 ∈ 𝓗_T. We seek m functions g_1m(t), …, g_mm(t) ∈ C¹[0, T] to construct the approximate solutions to problem (3.13)

   vm(x,t)=∑j=1mgjm(t)ωj(x),m=1,2,⋯ (3.14)

   satisfying

   ∫Ωvmt−Δvm−Δvmt−f(u)μdx=0,∫Ω−vm(0)dx=0,vm(0)=um0(x) (3.15)

   for any μ ∈ 𝓦_m and f(u) = ∣u∣^p−1u − ⨍_Ω ∣u∣^p−1udx. Taking μ = ω_j in (3.15) we have the following Cauchy problem of system of linear ordinary differential equations with the unknown g_jm(t)

   (1+λj)g˙jm(t)+λjgjm(t)=∫Ωf(u)ωjdx:=ξ(t),gjm(0)=∫Ωu0ωjdx. (3.16)

   Since ξ(t) ∈ C[0, T], for all j, according to standard existence theory for ordinary differential equations, the Cauchy problem (3.16) admits unique global solution g_jm(t) ∈ C¹[0, T]. Then the unique v_m(x, t) defined by (3.14) solves (3.15).

   Taking μ = v_mt in (3.15) and integrating the both sides over [0, t] ⊂ [0, T] with respect to t, we have

   ∫0t∥vmτ∥H12dτ+12∥∇vm∥22=12∥∇um0∥22+∫0t∫Ω|u|p−1uvmτdxdτ−∫0t∫Ω∫Ω−|u|p−1udxvmτdxdτ=12∥∇um0∥22+∫0t∫Ω|u|p−1uvmτdxdτ−∫0t∫Ω−|u|p−1udx∫Ωvmτdxdτ=12∥∇um0∥22+∫0t∫Ω|u|p−1uvmτdxdτ (3.17)

   by ∫_Ω v_m(0)dx = 0 and Lemma 2, i.e. ∫_Ω v_m dx = 0.

   For u ∈ 𝓗_T, i.e. ∥∇ u∥₂ ≤ R, we estimate the last term in the right hand side of (3.17) by Hölder, Sobolev and Young’s inequalities as follows

   ∫0t∫Ω|u|p−1uvmτdxdτ≤∫0t∫Ω|u|p|vmτ|dxdτ≤∫0t∫Ω|u|2npn+2dxn+22n∫Ω|vmτ|2nn−2dxn−22ndτ=∫0t∥u∥2npn+2p∥vmτ∥2nn−2dτ≤C∫0t∥u∥H1p∥vmτ∥H1dτ≤C∫0T∥u∥H12pdτ+12∫0t∥vmτ∥H12dτ≤CTR2p+12∫0t∥vmτ∥H12dτ. (3.18)

   Then combining (3.17), (3.18) and u_m0 ∈ 𝓗_T, we obtain

   ∫0t∥vmτ∥H12dτ+∥∇vm∥22≤C (3.19)

   for every m ≥ 1, where C is independent of m. Therefore, we see that the sequence {v_m} is bounded in L^∞([0, T]; H¹(Ω)) and {v_mt} is bounded in L²([0, T]; H¹(Ω)).

   Consequently, there exists a v and a subsequence {v_k} of {v_m} such that

   vk→vinL∞([0,T];H1(Ω))weak star,vkt→vtinL2([0,T];H1(Ω))weakly.

   In (3.15) letting m = k → ∞, we conclude the existence of a weak solution v of (3.13) with the above regularity. Moreover, observing (3.19) we get v ∈ H¹([0, T]; H¹(Ω)), then v ∈ C([0, T]; H¹(Ω)).

3. Arguing by contradiction, if v₁ and v₂ are two weak solutions of (3.13) with the same initial datum, by subtracting the two equations corresponding to v₁ and v₂ respectively, and testing it with v_1t − v_2t, we get

   ∫0t∥v1τ−v2τ∥H12dτ+12∥∇v1−∇v2∥22=0, (3.20)

   which implies v₁ ≡ v₂ in H¹(Ω). Then the uniqueness is proved.

4. By Step I, for any u ∈ 𝓗_T and the unique solution v to problem (3.13) we can define v := Φ(u). We claim that Φ(𝓗_T) ⊂ 𝓗_T is a contractive map. First for given u ∈ 𝓗_T, similar to (3.17), we have

   ∫0t∥vτ∥H12dτ+12∥∇v∥22=12∥∇u0∥22+∫0t∫Ω|u|p−1uvτdxdτ. (3.21)

   For the last term in above equality, by similar estimates as (3.18) we get

   ∫0t∫Ω|u|p−1uvτdxdτ≤CTR2p+∫0t∥vτ∥H12dτ (3.22)

   for all t ∈ (0, T]. Combining (3.21) and (3.22) gives

   ∥v∥H2≤12R2+CTR2p.

   For sufficiently small T, we see ∥ v∥_𝓗 ≤ R, which implies that Φ(𝓗_T) ⊂ 𝓗_T. Next we prove such map is contractive. Taking u₁, u₂ ∈ 𝓗_T to be the known functions in the linear terms of (3.13) respectively, subtracting the two equations in form of (3.13) for v₁ = Φ(u₁) and v₂ = Φ(u₂) respectively, setting v = v₁ − v₂ and testing the both sides by v_t, we can arrive at

   ∫0t∥vτ∥H12dτ+12∥∇v∥22=∫0t∫Ω|u1|p−1u1−|u2|p−1u2vτdxdτ. (3.23)

   For the last term of (3.23), by Lemma 4, Hölder, Sobolev and Young’s inequalities, we get

   ∫0t∫Ω|u1|p−1u1−|u2|p−1u2vτdxdτ≤∫0t∫Ωp(|u1|+|u2|)p−1|u1−u2|vτdxdτ≤C∫0t∥(|u1|+|u2|)p−1∥A1∥u1−u2∥A2∥vτ∥A3dτ=C∫0t∥|u1|+|u2|∥(p−1)A1p−1∥u1−u2∥A2∥vτ∥A3dτ≤C∫0t∥|u1|+|u2|∥H1p−1∥u1−u2∥H1∥vτ∥H1dτ≤C∫0T∥|u1|+|u2|∥H12(p−1)∥u1−u2∥H12dτ+∫0t∥vτ∥H12dτ≤CR2p−2T∥u1−u2∥H12+∫0t∥vτ∥H12dτ, (3.24)

   where (p−1)A1=2nn−2 and A2=A3=4n3n−(n−2)p−2<2nn−2 by (H). Combining (3.23) and (3.24), for any t ∈ [0, T] we obtain

   ∥∇v∥22≤CR2p−2T∥u1−u2∥H12,

   which by (3.12) gives

   ∥Φ(u1)−Φ(u2)∥H2=∥v∥H2=maxt∈[0,T]∥∇v∥22≤CR2p−2T∥u1−u2∥H12≤CR2p−2T∥∇(u1−u2)∥22≤CR2p−2T∥u1−u2∥H2:=δT∥u1−u2∥H2

   for some δ_T = CR^2p−2T < 1 as long as T is sufficiently small. Therefore, the map v = Φ(u) is contractive. By the Contraction Mapping Principle, there exists a unique weak local solution to (1.1) defined on [0, T].

5. By the arguments above, especially the requirements on δ_T, the sufficient conditions ensuring the contraction indicate that the existence time of the local solution only depends on the scale of the norm of the initial data by the connection between R and ∥∇ u₀∥₂. Therefore, the local solution u is continued as long as ∥ u∥_𝓗 remains bounded, see also Theorem 1 in [38] and Theorem 3.1 in [39] for similar argument. Hence, if the maximum existence time of local solution is T_max < ∞, we have

   limt→Tmax∥∇u∥22=∞. (3.25)

   By (2.5) and (2.8) we deduce

   12∥∇u∥22≤1p+1∥u∥p+1p+1+J(u0)for allt∈[0,Tmax),

   which together with the following Gagliardo-Nirenberg interpolation inequality

   ∥u∥B1≤C∥∇u∥B2α∥u∥B31−αfor1B1=1B2−1nα+1−αB3and0<α<1

   gives

   12∥∇u∥22−J(u0)≤1p+1∥u∥p+1p+1≤C∥∇u∥2(p+1)α∥u∥s(p+1)(1−α), (3.26)

   where B₁ = p + 1, B₂ = 2 and B₃ = s, and n(p−1)2 < s < p + 1 in order to ensure 0<α=2n(p+1−s)(p+1)(2n+2s−ns)<2p+1. Then

   12∥∇u∥22−(p+1)α−J(u0)∥∇u∥2−(p+1)α≤C∥u∥s(p+1)(1−α), (3.27)

   which together with (3.25) implies that

   limt→Tmax∥u∥s=∞.

   Combining Step I, Step II and Step III we complete the proof of Theorem 8.□

Lemma 9

(Invariant sets for J(u₀) < d). Let p satisfy (H), u₀ ∈ H¹(Ω), T be the maximal existence time of solution, then

1. the solution u of (1.1) with J(u₀) < d belongs to W for 0 ≤ t < T, provided I(u₀) > 0;

2. the solution u of (1.1) with J(u₀) < d belongs to V for 0 ≤ t < T, provided I(u₀) < 0.

Proof

1. Arguing by contradiction, by the continuity of I(u) respect to t, we suppose that t₀ ∈ (0, T) is the first time such that u(t₀) ∈ ∂ W, that is I(u(t₀)) = 0, ∥ u(t₀)∥_H¹ ≠ 0 or J(u(t₀)) = d. From

   ∫0t∥uτ∥H12dτ+J(u)=J(u0)<d,0≤t<T, (4.28)

   we know that J(u(t₀)) ≠ d. If I(u(t₀)) = 0, ∥ u(t₀)∥_H¹ ≠ 0, then by the definition of d we have J(u(t₀)) ≥ d, which contradicts (4.28).

2. The proof is similar to (i).

Next, we prove the global existence and uniqueness of solution with J(u₀) < d, also show the asymptotic behavior of global solution.

Theorem 10

(Global existence, uniqueness and asymptotic behavior for J(u₀) < d). Supposed that p satisfies (H). If u₀ ∈ H¹(Ω) and u₀ ∈ W, then problem (1.1) admits a unique global weak solution u ∈ L^∞(0, ∞; H¹(Ω)) with u_t ∈ L²(0, ∞; H¹(Ω)) which does not vanish in finite time and there exist a constant δ > 0 such that ∥u∥H12<∥u0∥H12e−2δt.

Proof

By Theorem 8, we know that the problem (1.1) admits a unique local solution u ∈ C([0, T]; H¹(Ω)). Next we will prove the existence time T of solution u(t) is infinite with u₀ ∈ W.

1. Assume {ω_j(x)} is an orthogonal basis of H¹(Ω). We give the construction of approximate solutions to problem (1.1)

   um(x,t)=∑j=1mgjm(t)ωj(x),m=1,2,⋯

   satisfying

   (umt,ωs)+(∇umt,∇ωs)+(∇um,∇ωs)=(f(um),ωs),s=1,2,⋯,m (4.29)

   and

   um(x,0)=∑j=1majmωj(x)→u0(x)inH1(Ω), (4.30)

   where

   f(u)=|u|p−1u−∫Ω−|u|p−1udxandajm=gjm(0).

   Multiplying (4.29) by g′_sm(t), integrating over [0, t] and summing for s gives

   ∫0t∥umτ∥H12dτ+J(um)=J(um(0)),0≤t<∞. (4.31)

   By (4.30) we know that J(u_m(0)) → J(u₀) < d and I(u_m(0)) → I(u₀) > 0, which combining (4.31) implies that

   ∫0t∥umτ∥H12dτ+J(um)<d,0≤t<∞, (4.32)

   and by Lemma 9, for 0 ≤ t < ∞ and sufficiently large m we have u_m(t) ∈ W. Combining (4.32) and

   J(um)=p−12(p+1)∥∇um∥22+1p+1I(um),

   gives

   ∫0t∥umτ∥H12dτ+p−12(p+1)∥∇um∥22<d,0≤t<∞

   for sufficiently large m, which yields a priori estimate

   ∫0t∥umτ∥H12dτ<d,0≤t<∞, (4.33)

   ∥∇um∥22<2(p+1)p−1d,0≤t<∞, (4.34)

   ∥ump∥qq=∥um∥p+1p+1≤C∗p+12(p+1)p−1dp+12,q=p+1p,0≤t<∞. (4.35)

   Therefore, there exists a u and a subsequence {u_ν} of {u_m} such that

   uν→uinL∞(0,∞;H1(Ω))weak star and a.e. inQ=Ω×(0,∞),uνp→upinL∞(0,∞;Lq(Ω))weak star,uνt→utinL2(0,∞;H1(Ω))weakly,|uν|p−1uν→|u|p−1uinL∞(0,∞;Lp+1p(Ω))weak star.

   In (4.29) we fixed s, letting m = ν → ∞, we conclude that

   (ut,ωs)+(∇ut,∇ωs)+(∇u,∇ωs)=(f(u),ωs)for alls,

   and

   (ut,v)+(∇ut,∇v)+(∇u,∇v)=(f(u),v)for allv∈H1(Ω),t∈(0,∞).

   Moreover, (4.30) gives u(x, 0) = u₀(x) in H¹(Ω).

   Assume that there exists a t₀ > 0 such that u(t₀) = 0. Then we get I(u(t₀)) = 0, which is contradictory to I(u) > 0. Thus the solution does not vanish in finite time.

2. If u₁ and u₂ are two solutions of (1.1) with the same initial data, then functions u₁ and u₂ satisfy the following equations for any ϕ ∈ L²(0, ∞; H¹(Ω))

   (u1t,ϕ)+(∇u1t,∇ϕ)+(∇u1,∇ϕ)=(f(u1),ϕ) (4.36)

   and

   (u2t,ϕ)+(∇u2t,∇ϕ)+(∇u2,∇ϕ)=(f(u2),ϕ), (4.37)

   where f(u) = ∣u∣^p−1u − ⨍_Ω ∣u∣^p−1udx. Let v = u₁ − u₂. By subtracting the above two equations, take ϕ = v in (4.36) and (4.37), integrating on (0, t), then

   ∫0t∫Ω(vτv+∇vτ∇v+|∇v|2)dxdτ=∫0t∫Ω|u1|p−1u1−|u2|p−1u2vdxdτ. (4.38)

   By Lemma 4, (3.24) and the boundedness of the solutions, we reach the inequality

   ∫0t∫Ω|u1|p−1u1−|u2|p−1u2vdxdτ≤∫0t∫Ωp(|u1|+|u2|)p−1|u1−u2|vdxdτ≤C∫0t∥(|u1|+|u2|)p−1∥A1∥u1−u2∥A2∥v∥A3dτ=C∫0t∥|u1|+|u2|∥(p−1)A1p−1∥u1−u2∥A2∥v∥A3dτ≤C∫0t∥|u1|+|u2|∥H1p−1∥u1−u2∥H1∥v∥H1dτ≤C∫0t∥v∥H12dτ, (4.39)

   where (p−1)A1=2nn−2 and A2=A3=4n3n−(n−2)p−2<2nn−2 by (H). As v(x, 0) = 0, both (4.38) and (4.39) give

   ∫0t∫Ω(vτv+∇vτ∇v)dxdτ=12∫Ω(v2(τ)|v(0)v(t)+|∇v(τ)|2|v(0)v(t))dx=12∫Ω(v2(t)+|∇v(t)|2)dx=12∥v∥H12≤C∫0t∥v∥H12dτ, (4.40)

   i.e.

   ∥v∥H12≤C∫0t∥v∥H12dτ.

   By Gronwall’s inequality, we get

   ∥v∥H12≤0⋅exp∫0tCds=0,

   that is ∥v∥H12=∥u1−u2∥H12=0. Thus u₁ = u₂ = 0 a.e. in Ω × (0, ∞).

3. Setting φ = u in (2.7) and noting that

   ∫0t∫Ω∫Ω−|u|p−1udxudxdτ=∫0t∫Ω−|u|p−1udx∫Ωudxdτ=0,

   we have

   12∥u∥H12−12∥u0∥H12+∫0t∥∇u∥22−∥u∥p+1p+1ds=0,

   that is

   12ddt∥u∥H12=−I(u). (4.41)

   According to 0 < J(u₀) < d, I(u₀) > 0 and Lemma 9, we have I(u) > 0 for 0 < t < ∞. Then by (2.9) and the norm ∥ u∥_H¹ is equivalent to the norm ∥∇ u∥₂ on H¹(Ω), there holds

   12ddt∥u∥H12=−I(u)=−∥∇u∥22+∥u∥p+1p+1≤−∥∇u∥22+C∗p+1∥∇u∥2p+1=C∗p+1∥∇u∥2p−1−1∥∇u∥22≤CC∗p+1∥∇u∥2p−1−1∥u∥H12, (4.42)

   and we shall show that C∗p+1∥∇u∥2p−1−1<0. By (2.5), (2.6), (2.8) and I(u) > 0 we deduce that

   J(u)>p−12(p+1)∥∇u∥22,

   that is

   ∥∇u∥22<2(p+1)p−1J(u)≤2(p+1)p−1J(u0). (4.43)

   From (4.43) and (2.10), we reach the following inequality

   C∗p+1∥∇u∥2p−1<C∗p+12(p+1)p−1J(u0)p−12<1. (4.44)

   From (4.42) there holds

   12ddt∥u∥H12<C(C∗p+1(2(p+1)p−1J(u0))p−12−1)∥u∥H12.

   Note that (4.44), we can see that there exists δ > 0 such that

   −δ:=C(C∗p+1(2(p+1)p−1J(u0))p−12−1).

   By Gronwall’s inequality, we have for δ > 0

   ∥u∥H12<∥u0∥H12e−2δt,0≤t<∞.

   The proof of this theorem is complete.□

   Here, we show a relationship between ∥∇u∥22 and the depth of potential well d with the initial value condition u₀ ∈ V, which is helpful for the proof of finite time blowup of solution.