On a strongly damped semilinear wave equation with time-varying source and singular dissipation

Definition 2.1

(Weak solution) u ( x , t ) is called a weak solution of (1.1)–(1.3) on Ω × [ 0 , T ] if u ∈ C ( [ 0 , T ] ; H 0 1 ( Ω ) ) with u t ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) and u t t ∈ L 2 ( 0 , T ; H − 1 ( Ω ) ) satisfying

1. for any η ∈ H 0 1 ( Ω ) and a.e. t ∈ ( 0 , T ) ,

   (2.2) ⟨ u t t , η ⟩ + ( ∇ u , ∇ η ) + ( ∇ u t , ∇ η ) + ( ∣ x ∣ − s u t , η ) = k ( t ) ( ∣ u ∣ p − 2 u , η )

   and ∫ 0 t u τ ∣ x ∣ s 2 2 2 d τ < + ∞ .

2. u ( x , 0 ) = u 0 ( x ) in H 0 1 ( Ω ) ; u t ( x , 0 ) = u 1 ( x ) in L 2 ( Ω ) .

Definition 2.2

(Finite time blow-up) Let u ( x , t ) be a weak solution of problems (1.1)–(1.3). We say that u ( x , t ) blows up in finite time if the maximal existence time satisfying T max < + ∞ and

Lemma 2.1

(Hardy-Sobolev inequality) (cf. [2]) Let R N = R k × R N − k , 2 ≤ k ≤ N , and x = ( y , z ) ∈ R N = R k × R N − k . For given p , s satisfying 1 < p < N , 0 ≤ s ≤ p , s < k , and m ( s , N , p ) = p ( N − s ) N − p , there exists a positive constant H = H ( s , N , p , k ) such that for any u ∈ W 0 1 , n ( R N ) ,

Remark 2.1

When m = p = s , this inequality is the classical Hardy inequality. With above inequality, we have u ∣ x ∣ s 2 2 2 ≤ H ‖ ∇ u ‖ 2 2 ( H > 0 ) . In fact, let m = 2 in (2.3), and by m ( s , N , p ) = p ( N − s ) N − p , we have p = 2 N N − s + 2 . For ∀ u ∈ H 0 1 ( Ω ) , by extending u ( x ) with u ( x ) = 0 in x ∈ R N ⧹ Ω , we have u ∈ H 1 ( R N ) , and (2.3) becomes

Due to 0 ≤ s ≤ 2 , then 2 N N + 2 ≤ γ ≤ 2 . We define γ ≔ 2 N N − s + 2 , then N − s + 2 N = 2 γ . By N > 2 , we know 2 N N + 2 > 1 , and then 1 < γ ≤ 2 . By (2.4) and L 2 ( Ω ) ↪ L γ ( Ω ) , we obtain

Lemma 2.2

Let u ∈ H 0 1 ( Ω ) ⧹ { 0 } for any t ∈ [ 0 , + ∞ ) , and the potential depth d ( t ) satisfies

where B 0 is the optimal embedding constant of H 0 1 ( Ω ) ↪ L p ( Ω ) , i.e.,

(ii) d ( t ) is positive nonincreasing, and there exists d ( ∞ ) ∈ [ 0 , d ( 0 ) ] such that d ( ∞ ) ≔ lim t → + ∞ d ( t ) .

Proof

(i) For any 0 ≠ u ∈ H 0 1 ( Ω ) and t ≥ 0 , let

By a simple computation, it can be known that there exists a unique λ ∗ ≔ ‖ ∇ u ‖ 2 2 k ( t ) ‖ u ‖ p p 1 p − 2 such that F ′ ( λ ∗ ) = 0 , F ( λ ) is strictly increasing on ( 0 , λ ∗ ) and strictly decreasing on ( λ ∗ , + ∞ ) . Hence, using the definition of the potential depth d and combining (2.6), we have

Obviously, it is known from the assumption of k ( t ) , k ( t ) ∈ C 1 ( [ 0 , + ∞ ) ) is a nondecreasing function satisfying k ( 0 ) > 0 , and for t ∈ [ 0 , + ∞ ) , well depth d ( t ) > 0 .

(ii) Based on (2.5) and the assumption of k ( t ) , it is easy to verify that d ( t ) is nondecreasing and d ( ∞ ) ∈ [ 0 , d ( 0 ) ] . The proof of Lemma 2.2 is completed.□

Lemma 2.3

Let u be a solution of problems (1.1)–(1.3), then E ( t ) is nonincreasing with respect to t.

Proof

Multiplying (1.1) by u t , then integrating over Ω × [ 0 , t ] , we arrive at

Utilizing the definition of E ( t ) , which can lead to

and with the assumption of k ( t ) , we have

The proof is completed.□

Lemma 3.1

For some T > 0 and every initial data ( ( u 0 ( x ) , u 1 ( x ) ) ) ∈ ℋ and every known u n satisfying

there exists a unique solution

which solves the linear problem

Proof

The standard approach to show this result is the Galerkin method.

Step 1: Approximate problem.

For ∀ j ∈ N , we choose a complete smooth orthogonal basis { ω j ( x ) } j = 1 + ∞ with ‖ ω j ‖ 2 = 1 in H 0 1 ( Ω ) , which is also orthogonal in L 2 ( Ω ) . We define finite dimensional space W h ≔ span { ω 1 , … , ω h } , ∀ h ∈ N , and we construct the approximate solution v n h ( t ) of problem (3.1):

admitting to the following Cauchy problem:

and

where v ¨ n h ( t ) and v ˙ n h ( t ) are derivatives of time variables t .

According to the standard theory for ordinary differential equations, for every j and n , there exists a unique global solution γ n j h ∈ C ( [ 0 , T ] ) of the aforementioned Cauchy problem. Therefore, the unique solution v n h solves (3.3). Next, we will prove that { v n h } has convergent subsequences, and its limit v n is the unique weak solution of problem (3.1).

Step 2: A priori estimates.

In (3.3), taking η = v ˙ n h ( t ) and integrating from 0 to t , we obtain

Next we estimate the last term in (3.4). By Hölder’s inequality and the assumption of k ( t ) , we derive

So next we estimate the term ‖ v ˙ n h ‖ 2 N N − 2 in the aforementioned formula. Since H 0 1 ( Ω ) ↪ L 2 N N − 2 ( Ω ) , we have ‖ v ˙ n h ‖ 2 N N − 2 ≤ C ‖ ∇ v ˙ n h ‖ 2 . Meanwhile, due to 2 N ( p − 1 ) N + 2 < 2 N N − 2 , it follows that the embedding H 0 1 ( Ω ) ↪ L 2 N N − 2 ( Ω ) ↪ L 2 N ( p − 1 ) N + 2 ( Ω ) and the estimate:

According to the aforementioned estimates, (3.5), and Young’s inequality, (3.4) becomes

where C ¯ T = 1 2 ‖ u 1 h ( x ) ‖ 2 2 + 1 2 ‖ ∇ u 0 h ( x ) ‖ 2 2 + 1 2 ( C T C 1 T ) 2 T and C ¯ T > 0 is independent of h for every h ≥ 1 . By this uniform estimate, we have

Therefore, we obtain { v ˙ n h } is uniformly bounded in L ∞ ( 0 , T ; L 2 ( Ω ) ) , { v n h } is uniformly bounded in L ∞ ( 0 , T ; H 0 1 ( Ω ) ) , and { ρ n ( x ) v ˙ n h } and { ∇ v ˙ n h } are uniformly bounded in L 2 ( 0 , T ; L 2 ( Ω ) ) . Further, since

which implies { v ˙ n h } is uniformly bounded in L 2 ( 0 , T ; H 0 1 ( Ω ) ) .

Step 3: Pass to the limit.

It follows from the aforementioned priori estimates, that there exist function v n h and a subsequence of { v n h } h = 1 ∞ , which we still denote by { v n h } h = 1 ∞ for convenience, such that