Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities

Theorem 1.1

Let Ω ⊂ R 2 be a bounded domain with smooth boundary. Assume that the initial data satisfy the conditions given in (1.2) and the parameters are such that χ i ∈ R , μ i > 0 , and a i > 0 for i = 1 , 2 . If m > 0 and k > 0 , then problem (1.1) has a global classical solution. Moreover, there exists C > 0 independent of t such that

Theorem 1.2

Let Ω ⊂ R 3 be a bounded domain with smooth boundary. Assume that the initial data satisfy the conditions given in (1.2) and the parameters are such that χ i > 0 , μ i > 0 , and a i > 0 for i = 1 , 2 . If m > 1 2 and 0 < k < 1 , then problem (1.1) has a global classical solution. Moreover, there exists C > 0 independent of t such that

Remark 1.1

In fact, the above results on the global existence of solutions can be extended to model (1.3). Specifically, when D ( u ) ≡ ( u + 1 ) m , S ( u ) ≡ u , f ( u ) ≡ r u − u 2 , and χ ( v ) ≡ v k , if m > 0 and k > 0 , the global existence of classical solutions can be established in two dimensions. Furthermore, in three dimensions, if m > 1 2 and 0 < k < 1 , the global existence of classical solutions can also be obtained.

Theorem 1.3

Assume that the conditions in either Theorem 1.1 or Theorem 1.2 hold. If one of the following conditions

1. 0 < k < m + 2 2 , α 1 ∈ ( 0 , 1 ) , α 2 ∈ ( 0 , 1 ) , and μ 1 = β μ 2 (for any given β > 0 ) in n = 2 ,

2. 0 < k < min { − 8 + 64 + 48 ( 2 m − 1 ) m + 5 3 ( m + 2 ) 24 , m + 2 2 } , α 1 ∈ ( 0 , 1 ) , α 2 ∈ ( 0 , 1 ) , and μ 1 = β μ 2 (for any given β > 0 ) in n = 3 ,

Here μ 1 * and μ 2 * denote constants depending on m , χ i , a i ( i = 1 , 2 ) , ∣ Ω ∣ , ‖ u 0 ‖ L ∞ , ‖ v 0 ‖ L ∞ , and ‖ w 0 ‖ W 1 , p . For details, refer the proof of Lemma 4.1.

Theorem 1.4

Assume that the conditions in either Theorem 1.1 or Theorem 1.2 hold. If one of the following conditions

1. 0 < k < m + 2 2 , α 1 ∈ ( 0 , 1 ) , α 2 ∈ ( 0 , 1 ) , and μ 1 = β μ 2 (for any given β > 0 ) in n = 2 ,

2. 0 < k < min { − 8 + 64 + 48 ( 2 m − 1 ) m + 5 3 ( m + 2 ) 24 , m + 2 2 } , α 1 ∈ ( 0 , 1 ) , α 2 ∈ ( 0 , 1 ) , and μ 1 = β μ 2 (for any given β > 0 ) in n = 3 ,

Here μ 2 ⋆ is a constant depending on m , χ i , a i ( i = 1 , 2 ) , ∣ Ω ∣ , ‖ u 0 ‖ L ∞ , ‖ v 0 ‖ L ∞ , and ‖ w 0 ‖ W 1 , p . For details, refer the proof of Lemma 4.3.

Lemma 2.1

(Local existence) Let Ω ⊂ R n ( n = 2 , 3 ) be a bounded domain with smooth boundary. Assume (1.2), m ≥ 0 , and k ≥ 0 hold, then there exists a constant T max ∈ ( 0 , ∞ ] such that problem (1.1) has a unique classical solution ( u , v , w )

satisfying u , v , w > 0 for all t > 0 , where T max denotes the maximal existence time. Moreover if T max < ∞ , then

Proof

Let h = ( u , v , w ) . Problem (1.1) can be reformulated as

If u , v ≥ 0 , then tr A = ( u + 1 ) m + ( v + 1 ) m + 1 > 0 and det A = ( u + 1 ) m ( v + 1 ) m > 0 . The continuity of the trace and determinant ensures the existence of an open neighborhood D 0 of [ 0 , ∞ ) 3 in R 3 such that, for all u , v , w ∈ D 0 , all the eigenvalues of A have positive real parts. Thus, defining the operators A , ℬ by A ( η ) h ≔ ∇ ⋅ ( A ( η ) ∇ h ) and ℬ ( η ) h ≔ ν ⋅ A ( η ) ∇ h for η ∈ D 0 , we see that ( A ( η ) , ℬ ( η ) ) are of separated divergence form and hence, normally elliptic for all η in D 0 (cf. [3, Example 4.3(e)]).

Hence, it follows from [3, Theorems 14.4 and 14.6] that for a given h 0 ∈ ( W 1 , p ( Ω ) ) 3 with p > n , there exists a maximal existence time T max > 0 and a unique h ∈ ( C 0 ( Ω ¯ × [ 0 , T max ) ) ) ∩ C 2,1 ( Ω ¯ × ( 0 , T max ) ) 3 that is a classical solution to (2.1). Moreover, the positivity of ( u , v , w ) can be derived through the maximum principle. Extensibility criterion is obtained from [3, Theorem 15.5]. Consequently, the proof is completed.□

Lemma 2.2

([32] Lemma 2.2) Let T > 0 , τ ∈ ( 0 , T ) , a > 0 , and b > 0 . Suppose that y : [ 0 , T ) → [ 0 , ∞ ) is absolutely continuous and fulfills

with some nonnegative function h ∈ L loc 1 ( [ 0 , T ) ) satisfying ∫ t t + τ h ( t ) ≤ b for all t ∈ [ 0 , T − τ ) . Then,

Lemma 2.3

[18] Let p ≥ 1 and q ≥ 1 satisfy

Assuming g 0 ∈ W 1 , q ( Ω ) and g is a classical solution of the following equations:

for some T ∈ ( 0 , ∞ ] . If f ∈ L ∞ ( ( 0 , T ) ; L p ( Ω ) ) , then g ∈ L ∞ ( ( 0 , T ) ; W 1 , q ( Ω ) ) .

Lemma 2.4

Let ( e t d Δ ) t ≥ 0 be the Neumann heat semigroup in Ω , and let λ 1 > 0 denote the first nonzero eigenvalue of − d Δ in Ω under Neumann boundary conditions. Then, for all t > 0 , there exist some constants γ i ( i = 1 , 2 , 3 , 4 ) depending only on Ω such that

1. If 2 ≤ p < ∞ , then

   (2.2) ‖ ∇ e t d Δ z ‖ L p ≤ γ 1 e − λ 1 d t ‖ ∇ z ‖ L p ,

   for all z ∈ W 1 , p ( Ω ) .

2. If 1 ≤ q ≤ p ≤ ∞ , then

   (2.3) ‖ ∇ e t d Δ z ‖ L p ≤ γ 2 1 + t − 1 2 − n 2 1 q − 1 p e − λ 1 d t ‖ z ‖ L q ,

   for all z ∈ L q ( Ω ) .

3. If 1 ≤ q ≤ p ≤ ∞ , then

   (2.4) ‖ e t d Δ z ‖ L p ≤ γ 3 1 + t − n 2 1 q − 1 p e − λ 1 d t ‖ z ‖ L q ,

   for all z ∈ L q ( Ω ) .

4. If 1 < q ≤ p < ∞ or 1 < q < ∞ and p = ∞ , then

   (2.5) ‖ e t d Δ ∇ ⋅ z ‖ L p ≤ γ 4 1 + t − 1 2 − n 2 1 q − 1 p e − λ 1 d t ‖ z ‖ L q ,

   for all z ∈ ( L q ( Ω ) ) n .

Lemma 2.5

[23] Assume that Ω is a bounded domain, and let g ∈ C 2 ( Ω ) satisfy ∂ g ∂ v = 0 on ∂ Ω . Then, we have

where κ = κ ( Ω ) is an upper bound of the curvatures of ∂ Ω .

Lemma 2.6

Assume the assumptions in Lemma 2.1 hold, then there exist constants K 1 > 0 and K 2 > 0 independent of t such that

and

for all t ∈ ( 0 , T ˜ max ) , where

Proof

By straightforward computation using (1.1) and integration by parts, we can derive that

Cauchy-Schwarz’s inequality entails that

and

which together with comparison principle for ordinary differential equation give

and

We can therefore integrate (2.8) and (2.9) over ( t , t + τ ) to obtain

and

On the other hand, by combining (2.12) and (2.13), we can obtain

Consequently, the proof is completed.□

Lemma 2.7

Assume the assumptions in Lemma 2.1 hold. Then, the solution of (1.1) satisfies

and

where K j ≔ K j ( m , μ 1 , μ 2 , ∣ Ω ∣ , ‖ u 0 ‖ L ∞ , ‖ v 0 ‖ L ∞ , ‖ w 0 ‖ W 1 , p ) > 0 ( j = 3 , 4 ) are constants independent of t.

Proof

Multiplying the third equation of (1.1) by − Δ w , integrating the result over Ω , we can use Young’s inequality to obtain

which gives

Finally, applying Lemma 2.2 and (2.7), we have

Thereupon, an integration of (2.17) over ( t , t + τ ) yields

Thus, the proof of Lemma 2.7 is completed.□