Open Access

Global behavior for a diffusive predator-prey system with Holling type II functional response

Boundary Value Problems20122012:111

DOI: 10.1186/1687-2770-2012-111

Received: 6 April 2012

Accepted: 25 September 2012

Published: 9 October 2012

Abstract

A strongly coupled self- and cross-diffusion predator-prey system with Holling type II functional response is considered. Using the energy estimate, Sobolev embedding theorem and bootstrap arguments, the global existence of non-negative classical solutions to this system in which the space dimension is not more than five is obtained.

MSC: 35K50, 35K55, 35K57, 92D40.

Keywords

predator-prey system cross-diffusion global solution

1 Introduction

In this paper, we consider the global existence of non-negative classical solutions to the following diffusion predator-prey system with Holling type II functional response:
{ u t = Δ ( d 1 u + α 11 u 2 ) + α u ( 1 u K ) β m u v 1 + a m u , x Ω , t > 0 , v t = Δ ( d 2 v + α 21 u v + α 22 v 2 ) r v + c β m u v 1 + a m u , x Ω , t > 0 , η u ( x , t ) = η v ( x , t ) = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) 0 , v ( x , 0 ) = v 0 ( x ) 0 , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ1_HTML.gif
(1.1)

where Ω is a bounded region in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq1_HTML.gif ( n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq2_HTML.gif) with a smooth boundary Ω; η is the outward normal on Ω, η = / η https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq3_HTML.gif; u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq4_HTML.gif and v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq5_HTML.gif are non-negative smooth functions and are not identically zero; u and v denote the population densities of predator and prey, respectively; α, β, r, a, K, m and c are positive constants, and m ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq6_HTML.gif; d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq7_HTML.gif and d 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq8_HTML.gif are the diffusion rates of the two species; α i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq9_HTML.gif ( i , j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq10_HTML.gif) are given non-negative constants, α 11 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq11_HTML.gif and α 22 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq12_HTML.gif are self-diffusion rates; α 21 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq13_HTML.gif is the cross-diffusion rate. It means that the diffusion is from one species of high-density areas to the other species of low-density areas. See [1, 2] for more details on the ecological backgrounds of this system.

Obviously, the non-negative equilibrium solutions of system (1.1) are ( K , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq14_HTML.gif and ( u , v ) = ( r m ( c β a r ) , α c K m ( c β a r ) α c r K m 2 ( c β a r ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq15_HTML.gif. For the reaction-diffusion problem of system (1.1), i.e., α i i = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq16_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq17_HTML.gif), the global attraction, persistence and stability of non-negative equilibrium solutions are studied in [3]. The main result can be summarized as follows:
  1. (1)

    If m < r K ( c β a r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq18_HTML.gif, a semi-trivial solution ( K , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq14_HTML.gif is globally asymptotically stable;

     
  2. (2)

    If r K ( c β a r ) < m r K ( c β a r ) + 1 K a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq19_HTML.gif and c β > a r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq20_HTML.gif, a unique positive constant solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq21_HTML.gif is globally asymptotically stable;

     
  3. (3)

    If r K ( c β a r ) < m < r K ( c β a r ) + c β K a ( c β a r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq22_HTML.gif and c β > a r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq20_HTML.gif, a positive constant solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq21_HTML.gif is locally asymptotically stable.

     

In view of the study of dynamic behavior of a predator-prey reaction-diffusion system with Holling type II functional response, a natural problem is what the global behavior for a predator-prey cross-diffusion system (1.1) is. To the best of our knowledge, the existing results are very few. In this paper, we consider the space dimension to be less than six, and initial function u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq4_HTML.gif and v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq5_HTML.gif under some smooth conditions, using the energy estimate, Sobolev embedding theorem and bootstrap arguments, we consider the global existence of non-negative classical solutions for system (1.1).

We denote Q T = Ω × ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq23_HTML.gif. u W q 2 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq24_HTML.gif means that u, u x i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq25_HTML.gif, u x i x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq26_HTML.gif ( i , j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq27_HTML.gif) and u t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq28_HTML.gif are in L q ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq29_HTML.gif. u L p , q ( Q T ) = [ 0 T ( Ω | u ( x , t ) | p d x ) q p d t ] 1 q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq30_HTML.gif. u V 2 ( Q T ) = sup 0 t T u ( , t ) L 2 ( Ω ) + u L 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq31_HTML.gif.

2 Auxiliary results

Lemma 2.1 Let ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq32_HTML.gif be the solution of (1.1). There exists a positive constant M 0 ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq33_HTML.gif such that
0 u M 0 , 0 v , t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ2_HTML.gif
(2.1)
Proof Firstly, the existence of local solutions for system (1.1) is given in [46]. Roughly speaking, if u 0 , v 0 W p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq34_HTML.gif, p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq35_HTML.gif, there exists the maximum T + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq36_HTML.gif such that system (1.1) admits a unique non-negative solution
u , v C ( [ 0 , T ) , W p 1 ( Ω ) ) C ( ( 0 , T ) , C ( Ω ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ3_HTML.gif
(2.2)
If
sup { u ( , t ) w p 1 ( Ω ) , v ( , t ) w p 1 ( Ω ) : 0 < t < T } < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equa_HTML.gif

then T = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq37_HTML.gif.

Choose M 0 = max { K , u 0 L ( Ω ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq38_HTML.gif. By use of the maximum principle, the non-negative solution of system (1.1) can be derived from the maximum principle, i.e., u , v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq39_HTML.gif for all t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq40_HTML.gif. This completes the proof of Lemma 2.1. □

Lemma 2.2 Let X = ( d 1 + α 11 u ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq41_HTML.gif, u L ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq42_HTML.gif for the solution to the following equation:
u t = Δ [ ( d 1 + α 11 u ) u ] + α u ( 1 u K ) β m u v 1 + a m u , ( x , t ) Ω × ( 0 , T ) , η u = 0 , ( x , t ) Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) 0 , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equb_HTML.gif
where d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq7_HTML.gif, α 11 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq11_HTML.gif are positive constants and 0 u L 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq43_HTML.gif. Then there exists a positive constant C ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq44_HTML.gif, depending on u 0 W 2 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq45_HTML.gif and u 0 L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq46_HTML.gif, such that
X W 2 2 , 1 ( Q T ) C ( T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ4_HTML.gif
(2.3)
Furthermore,
X V 2 ( Q T ) , u L 2 ( n + 2 ) n ( Q T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ5_HTML.gif
(2.4)
Proof From X = ( d 1 + α 11 u ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq41_HTML.gif, it is easy to find that
X t = ( d 1 + 2 α 11 u ) Δ X + C 1 C 2 v , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ6_HTML.gif
(2.5)
where C 1 = d 1 α u + ( 2 α 11 α d 1 α K ) u 2 2 α 11 α K u 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq47_HTML.gif and C 2 = β m u 1 + a m u ( d 1 + 2 α 11 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq48_HTML.gif. C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq49_HTML.gif and C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq50_HTML.gif are bounded in Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq51_HTML.gif from (2.1). Multiplying (2.5) by Δ X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq52_HTML.gif and integrating by parts over Q t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq53_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ7_HTML.gif
(2.6)
Using the Hölder inequality and Young inequality to estimate the right-hand side of (2.6), we have
C 1 + C 2 v L 2 ( Q T ) Δ X L 2 ( Q T ) m 1 ( 1 + v L 2 ( Q T ) ) Δ X L 2 ( Q T ) m 1 2 ( 1 + M 3 ) 2 2 d 1 + d 1 2 Δ X L 2 ( Q T ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ8_HTML.gif
(2.7)
with some m 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq54_HTML.gif. Substituting (2.7) into (2.6), we obtain
sup 0 t T Ω | X ( x , t ) | 2 d x + d 1 Q t | Δ X | 2 d x d t m 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equc_HTML.gif
where m 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq55_HTML.gif depends on u 0 W 2 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq45_HTML.gif and u 0 L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq46_HTML.gif. So, we know X V 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq56_HTML.gif. Since X L 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq57_HTML.gif, it follows from the elliptic regularity estimate [[7], Lemma 2.3] that
Q T | X x i x j | 2 d x d t m 3 , i , j = 1 , , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equd_HTML.gif

From (2.5), we have X t L 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq58_HTML.gif. Hence, X W 2 2 , 1 ( Q T ) C ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq59_HTML.gif. Moreover, (2.4) can be obtained by use of the Sobolev embedding theorem. □

Lemma 2.3 Assume that w W P 2 , 1 ( Q T ) C 2 , 1 ( Ω ¯ × [ 0 , T ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq60_HTML.gif is a bounded function satisfying
w t a ( x , t , w ) Δ w + f ( x , t ) in  Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Eque_HTML.gif

with the boundary condition w v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq61_HTML.gif on Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq62_HTML.gif, where f L P ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq63_HTML.gif. Then W is in L 2 p ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq64_HTML.gif.

The proof of the above lemma can be found in [[8], Proposition 2.1].

The following result can be derived from Lemma 2.3 and Lemma 2.4 of [9].

Lemma 2.4 Let p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq35_HTML.gif, p ˜ = 2 + 4 p n ( p + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq65_HTML.gif. If
sup 0 t T w L 2 p p + 1 ( Ω ) + w L 2 ( Q T ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equf_HTML.gif
and there exist positive constants β ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq66_HTML.gif and C T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq67_HTML.gif such that Ω | w ( , t ) | β d x C T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq68_HTML.gif ( t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq69_HTML.gif), there exists a positive constant M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq70_HTML.gif, independent of w but possibly depending on n, Ω, p, β and C T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq67_HTML.gif, such that
w L p ˜ ( Q T ) M { 1 + ( sup 0 t T w ( t ) L 2 p p + 1 ( Ω ) ) 4 p n ( p + 1 ) p ˜ w L 2 ( Q T ) 2 p ˜ } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equg_HTML.gif

Finally, one proposes some standard embedding results which are important to obtain the L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq71_HTML.gif and C 2 + α , 1 + α 2 ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq72_HTML.gif normal estimates of the solution for (1.1).

Lemma 2.5 There exists a constant C 3 ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq73_HTML.gif such that u L 4 ( Q T ) C 3 ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq74_HTML.gif.

Proof Let δ = α 11 / d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq75_HTML.gif, X = ( 1 + δ u ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq76_HTML.gif. By Lemma 2.1, u is bounded. Therefore, X is also bounded. By Lemma 2.2, we have X W 2 2 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq77_HTML.gif. Moreover, X satisfies
X t d 1 ( 1 + 2 δ u ) Δ X + α u ( 1 + 2 δ u ) = d 1 2 + 4 δ d 1 X Δ X + α u ( 1 + 2 δ u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equh_HTML.gif

By Lemma 2.3 with p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq78_HTML.gif, a ( x , t , ξ ) = d 1 2 + 4 δ d 1 ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq79_HTML.gif, f ( x , t ) = α u ( x , t ) ( 1 + 2 δ u ( x , t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq80_HTML.gif, we obtain the desired result. □

Lemma 2.6 Let Ω R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq81_HTML.gif be a fixed bounded domain and Ω C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq82_HTML.gif. Then for all u W q 2 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq83_HTML.gif with q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq84_HTML.gif, one has
  1. (1)

    u L p ( Q T ) C u W q 2 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq85_HTML.gif, 1 p ( n + 2 ) q n + 2 q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq86_HTML.gif, q < n + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq87_HTML.gif;

     
  2. (2)

    u L p ( Q T ) C u W q 2 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq85_HTML.gif, 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq88_HTML.gif, q = n + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq89_HTML.gif;

     
  3. (3)

    u C α , α 2 ( Q T ) C u W q 2 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq90_HTML.gif, 1 n + 2 q α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq91_HTML.gif, q > n + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq92_HTML.gif,

     

where C is a positive constant dependent on q, n, Ω, T.

3 The existence of classical solutions

The main result about the global existence of non-negative classical solutions for the cross-diffusion system (1.1) is given as follows.

Theorem 3.1 Assume that u 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq93_HTML.gif and v 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq94_HTML.gif satisfy homogeneous Neumann boundary conditions and belong to C 2 + α ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq95_HTML.gif for some α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq96_HTML.gif. Then system (1.1) has a unique non-negative solution ( u , v ) C 2 + α , 1 + α 2 ( Ω ¯ × [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq97_HTML.gif if the space dimension is n 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq98_HTML.gif.

Proof When n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq99_HTML.gif, the proof is similar to the methods of [1012]. So, we just give the proof of Theorem 3.1 for n = 2 , 3 , 4 , 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq100_HTML.gif. The proof is divided into three parts.
  1. (i)

    L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq101_HTML.gif-, L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq102_HTML.gif-estimate and L q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq103_HTML.gif-estimate for v.

     
Firstly, integrating the first equation of (1.1) over Ω, we have
d d t Ω u d x = Ω u ( α α u K β m v 1 + a m u ) d x Ω α u d x α K Ω u 2 d x α Ω u d x α K | Ω | ( Ω u d x ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equi_HTML.gif
Thus, for all t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq40_HTML.gif, we can obtain
Ω u d x M 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equj_HTML.gif

where M 1 = max { K | Ω | , Ω u 0 d x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq104_HTML.gif.

Furthermore,
u L 1 ( Q T ) 0 T M 1 d t M 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ9_HTML.gif
(3.1)
Secondly, linear combination of the second and first equations of (1.1) and integrating over Ω yields
d d t Ω ( c u + v ) d x = c Ω α u d x c α K Ω u 2 d x r Ω v d x c ( r + α ) Ω u d x c α K | Ω | ( Ω u d x ) 2 r Ω ( c u + v ) d x c ( r + α ) M 1 r Ω ( c u + v ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equk_HTML.gif
So, we get
Ω ( c u + v ) d x max { c ( r + α ) M 1 r , Ω ( c u 0 ( x ) + v 0 ( x ) ) d x } M 2 , t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ10_HTML.gif
(3.2)
Further,
v L 1 ( Q T ) 0 T M 1 d t M 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ11_HTML.gif
(3.3)
Then multiplying both sides of the second equation of system (1.1) by v and integrating over Ω, we obtain
1 2 d d t Ω v 2 d x = Ω v [ d 2 v + α 21 u v + α 22 v 2 ] d x r Ω v 2 d x + c m β Ω u v 2 1 + a m u d x d 2 Ω | v | 2 d x α 21 Ω v v u d x 2 α 22 Ω v | v | 2 d x + c β a Ω v 2 d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equl_HTML.gif
Integrating the above expression in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq105_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ12_HTML.gif
(3.4)
Estimating the first term on the right-hand side of (3.4),
| Q t u v v d x d t | = | Q t u v 1 2 v v 1 2 d x d t | ε 1 Q t v | u | 2 d x d t + 1 4 ε 1 Q t v | v | 2 d x d t ε 2 Q t v 2 d x d t + 1 4 ε 2 Q t | u | 4 d x d t + 1 4 ε 1 Q t v | v | 2 d x d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ13_HTML.gif
(3.5)
Substituting (3.5) into (3.4) yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ14_HTML.gif
(3.6)
Select ε 1 > α 21 8 α 22 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq106_HTML.gif and denote C 4 = 4 α 22 α 21 2 ε 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq107_HTML.gif. Notice the positive equilibrium point of (1.1) exists under condition c β > a r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq20_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ15_HTML.gif
(3.7)
By Lemma 2.5, u L 4 ( Q T ) C 3 ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq74_HTML.gif. Integrating the above inequality and using the Gronwall inequality, we get
sup 0 < t < T Ω v 2 d x C ( T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equm_HTML.gif
Hence, there exists a positive constant M 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq108_HTML.gif such that Ω v 2 d x M 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq109_HTML.gif. Furthermore, we have
v L 2 ( Q T ) 0 T M 2 d t M 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ16_HTML.gif
(3.8)
Secondly, multiplying both sides of the second equation of system (1.1) by q v q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq110_HTML.gif ( q > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq111_HTML.gif) and integrating over Ω, we have
d d t Ω v q d x 4 ( q 1 ) d 2 q Ω | ( v q 2 ) | 2 d x 8 q ( q 1 ) α 22 ( q + 1 ) 2 Ω | ( v q + 1 2 ) | 2 d x q ( q 1 ) α 21 Ω v q 1 u v d x + q Ω v q ( r + c β m u 1 + a m u ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equn_HTML.gif
Integrating the above equation over [ 0 , t ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq112_HTML.gif ( t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq113_HTML.gif), it is clear that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ17_HTML.gif
(3.9)
By Lemma 2.2, it can be found that u L 2 ( n + 2 ) n ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq114_HTML.gif. According to the Hölder inequality and Young inequality, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ18_HTML.gif
(3.10)
Choose an appropriate number ε satisfying C 3 ε 2 8 q ( q 1 ) α 22 ( q + 1 ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq115_HTML.gif. Substituting (3.10) into (3.9) and taking v ¯ = v q + 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq116_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ19_HTML.gif
(3.11)
Let
E sup 0 < t < T Ω v ¯ 2 q q + 1 ( x , t ) d x + Q T | v ¯ | 2 d x d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equo_HTML.gif
From (3.11), we know
E C 4 ( 1 + v ¯ L ( q 1 ) ( n + 2 ) q + 1 ( Q T ) 2 ( q 1 ) q + 1 + v ¯ L 2 q q + 1 ( Q T ) 2 q q + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equp_HTML.gif
When q < n ( n + 4 ) n 2 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq117_HTML.gif, it is easy to find that 2 q q + 1 < 2 < q ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq118_HTML.gif and ( q 1 ) ( n + 2 ) q + 1 < q ˜ = 2 + 4 q n ( q + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq119_HTML.gif. So,
E C 5 ( 1 + v ¯ L q ˜ ( Q T ) 2 ( q 1 ) q + 1 + v ¯ L q ˜ ( Q T ) 2 q q + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ20_HTML.gif
(3.12)
Set β = 2 q + 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq120_HTML.gif. It follows from the L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq121_HTML.gif-estimate for v that
v ¯ L β ( Ω ) = ( Ω | v ¯ ( , t ) | β d x ) 1 β = v L 1 ( Ω ) 1 β M 2 1 β , t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equq_HTML.gif
By Lemma 2.4 and (3.12), we know
E C 5 [ 1 + ( M + M sup 0 < t < T v ¯ ( , t ) L 2 q q + 1 ( Ω ) 4 q n ( q + 1 ) q ˜ v ¯ L 2 ( Q T ) 2 q ˜ ) 2 ( q 1 ) q + 1 + ( M + M sup 0 < t < T v ¯ ( , t ) L 2 q q + 1 ( Ω ) 4 q n ( q + 1 ) q ˜ v ¯ L 2 ( Q T ) 2 q ˜ ) 2 q q + 1 ] C 6 [ 1 + ( sup 0 < t < T v ¯ ( , t ) L 2 q q + 1 ( Ω ) 2 q q + 1 ) 4 ( q 1 ) n ( q + 1 ) q ˜ ( v ¯ L 2 ( Q T ) 2 ) 2 ( q 1 ) ( q + 1 ) q ˜ + ( sup 0 < t < T v ¯ ( , t ) L 2 q q + 1 ( Ω ) 2 q q + 1 ) 4 ( q 1 ) n ( q + 1 ) q ˜ ( v ¯ L 2 ( Q T ) 2 ) 2 q ( q + 1 ) q ˜ ] C 6 ( 1 + E 4 ( q 1 ) n ( q + 1 ) q ˜ E 2 ( q 1 ) ( q + 1 ) q ˜ + E 4 q n ( q + 1 ) q ˜ E 2 q ( q + 1 ) q ˜ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ21_HTML.gif
(3.13)
Obviously, 4 ( q 1 ) n ( q + 1 ) q ˜ + 2 ( q 1 ) ( q + 1 ) q ˜ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq122_HTML.gif and 4 q n ( q + 1 ) q ˜ + 2 q ( q + 1 ) q ˜ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq123_HTML.gif. It is easy to know that E is bounded by use of reduction to absurdity. Since q < n ( n + 4 ) n 2 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq117_HTML.gif, ( q + 1 ) q ˜ 2 ( 1 , 2 ( n + 1 ) n 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq124_HTML.gif. So, v ¯ L q ˜ ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq125_HTML.gif is bounded, i.e., v L ( q + 1 ) q ˜ 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq126_HTML.gif. Denote ( q + 1 ) q ˜ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq127_HTML.gif still as q. So,
v L q ( Q T ) , q ( 1 , 2 ( n + 1 ) n 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ22_HTML.gif
(3.14)
Finally, when n = 2 , 3 , 4 , 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq100_HTML.gif, ( n 2 4 ) q < n 2 + 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq128_HTML.gif with q = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq129_HTML.gif. For n 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq130_HTML.gif, taking q = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq129_HTML.gif in (3.9), it follows from (3.8) that there exists a positive constant M 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq131_HTML.gif such that
v V 2 ( Q T ) M 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ23_HTML.gif
(3.15)
By embedding theorem, we get
v L 2 ( n + 2 ) n ( Q T ) M 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equr_HTML.gif
  1. (ii)

    L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq71_HTML.gif-estimate for v.

     
The second equation of system (1.1) can be written as the following divergence form:
v t = i , j = 1 n x i ( a i j ( x , t ) v x j ) + i = 1 n x i ( a i ( x , t ) v ) + v ( r + c β m u 1 + a m u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ24_HTML.gif
(3.16)

where a i j ( x , t ) = ( d 2 + α 21 u + 2 α 22 v ) δ i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq132_HTML.gif, a i ( x , t ) = α 21 u x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq133_HTML.gif and δ i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq134_HTML.gif is the Kronecker sign.

In order to apply the maximum principle [13] to (3.16), we need to prove the following conditions:
  1. (1)

    v V 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq135_HTML.gif is bounded;

     
  2. (2)

    i , j = 1 n a i j ( x , t ) ξ i ξ j ν i = 1 n ξ i 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq136_HTML.gif;

     
  3. (3)

    i = 1 n a i 2 ( x , t ) , v ( r + c β m u 1 + a m u ) L p , r ( Q T ) μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq137_HTML.gif,

     
where ν, μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq138_HTML.gif are positive constants and
1 r + n 2 p = 1 χ , 0 < χ < 1 , p [ n 2 ( 1 χ ) , + ) , r [ 1 1 χ , + ) , n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ25_HTML.gif
(3.17)
Next, we will show that the above conditions (1) to (3) are satisfied for (3.16). When n 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq130_HTML.gif, it is easy to find that the condition (1) is satisfied by use of (3.15). Since i , j = 1 n a i j ( x , t ) ξ i ξ j d 3 | ξ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq139_HTML.gif for all ξ R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq140_HTML.gif, the condition (2) is verified. In view of the condition (3), we take appropriate q and r. Rewrite the first equation of system (1.1) as
u t = [ ( d 1 + 2 α 11 u ) u ] + α u ( 1 u K ) β m u v 1 + a m u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ26_HTML.gif
(3.18)
When n = 2 , 3 , 4 , 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq100_HTML.gif, n + 2 2 < 2 ( n + 1 ) n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq141_HTML.gif, it is clear that d 1 + 2 α 11 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq142_HTML.gif has an upper bound over Q ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq143_HTML.gif by Lemma 2.1. Set
q ( n + 2 2 , 2 ( n + 1 ) n 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equs_HTML.gif
From (3.14), we have α u ( 1 u K ) β m u v 1 + a m u L q ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq144_HTML.gif. Therefore, all conditions of the Hölder continuity theorem [[5], Theorem 10.1] hold for (3.18). Hence,
u C β , β 2 ( Q ¯ T ) , β ( 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ27_HTML.gif
(3.19)
We will discuss (2.5) which is the corresponding form of (3.18). It follows from (2.1) and (3.14) that C 1 C 2 v L q ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq145_HTML.gif, q ( n + 2 2 , 2 ( n + 1 ) n 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq146_HTML.gif. From (3.19), we obtain d 1 + α 11 u C β , β 2 ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq147_HTML.gif. Thus, according to the parabolic regularity result of [[5], pp.341-342, Theorem 9.1], we can conclude that
X W q 2 , 1 ( Q T ) , q ( n + 2 2 , 2 ( n + 1 ) n 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ28_HTML.gif
(3.20)

which implies that X L ( n + 2 ) q n + 2 q ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq148_HTML.gif by Lemma 2.6.

Since X = ( d 1 + 2 α 11 u ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq149_HTML.gif, we have u = ( d 1 + 2 α 11 u ) 1 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq150_HTML.gif, i.e., u L ( n + 2 ) q n + 2 q ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq151_HTML.gif. It means that | u | 2 , | v | 2 L ( n + 2 ) q 2 ( n + 2 q ) ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq152_HTML.gif. So, i = 1 n a i 2 ( x , t ) L ( n + 2 ) q 2 ( n + 2 q ) ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq153_HTML.gif. From (2.1) and (3.14), v ( r + c β m u 1 + a m u ) L q ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq154_HTML.gif.

Then the condition (3) and (3.17) are satisfied by choosing p = r = ( n + 2 ) p 2 ( n + 2 p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq155_HTML.gif. According to the maximum principle [[13], p.181, Theorem 7.1], we can conclude that v L ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq156_HTML.gif. From (2.1), there exists a positive constant M 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq157_HTML.gif such that
u L ( Q T ) , v L ( Q T ) M 5 , T > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ29_HTML.gif
(3.21)
Therefore, the global solution to the problem (1.1) exists.
  1. (iii)

    The existence of classical solutions.

     

Under the conditions of Theorem 3.1, we consider above global solutions ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq32_HTML.gif to be classical. By (3.20) and Lemma 2.6, we know X C α , α 2 ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq158_HTML.gif, α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq159_HTML.gif. It follows from Lemma 3.3 in [13] that X C 1 + α , 1 + α 2 ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq160_HTML.gif. Since X = ( d 1 + α 11 u ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq41_HTML.gif, we have u = d 1 d 1 2 + 4 α 11 X 2 α 11 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq161_HTML.gif.

So,
u C 1 + α , 1 + α 2 ( Q ¯ T ) , α ( 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ30_HTML.gif
(3.22)
Rewrite the second equation of system (1.1) as
v t = [ ( d 2 + α 21 u + 2 α 22 v ) v + α 21 u v ] + v ( r + c β m u 1 + a m u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equt_HTML.gif
Therefore, we can conclude that v ( r + c β m u 1 + a m u ) L ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq162_HTML.gif, u, v, u and v are all bounded. By the Schauder estimate [13], there exists α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq163_HTML.gif such that
v C α , α 2 ( Q ¯ T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ31_HTML.gif
(3.23)
Furthermore, by the Schauder estimate, we obtain
u C 2 + σ , 1 + σ 2 ( Q ¯ T ) , σ = min { α , α } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ32_HTML.gif
(3.24)
Next, the regularity of v will be discussed. Set v ¯ = ( d 2 + α 21 u + α 22 v ) v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq164_HTML.gif. So, v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq165_HTML.gif satisfies
v ¯ t = ( d 2 + α 21 u + 2 α 22 v ) Δ v ¯ + f ( x , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ33_HTML.gif
(3.25)
where f ( x , t ) = ( d 2 + α 21 u + 2 α 22 v ) v ( r + c β m u 1 + a m u ) + α 21 u t v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq166_HTML.gif. According to (3.22) to (3.24), we have d 2 + α 21 u + 2 α 22 v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq167_HTML.gif, f ( x , t ) C σ , σ 2 ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq168_HTML.gif. Applying the Schauder estimate to (3.25), we know
v ¯ C 2 + σ , 1 + σ 2 ( Q ¯ T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equu_HTML.gif
From v = ( d 3 + α 21 u ) + ( d 2 + α 21 u ) 2 + 4 α 22 v ¯ 2 α 22 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq169_HTML.gif, we can see
v C 2 + σ , 1 + σ 2 ( Q ¯ T ) , σ = min { α , α } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equ34_HTML.gif
(3.26)
Combining (3.24) and (3.26), we get
u , v C 2 + σ , 1 + σ 2 ( Q ¯ T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_Equv_HTML.gif

Therefore, the result of Theorem 3.1 can be obtained for α < α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq170_HTML.gif, namely σ = α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq171_HTML.gif. When α > α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq172_HTML.gif, namely σ < α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq173_HTML.gif, we have C 2 + σ , 1 + σ 2 ( Q ¯ T ) C α , α 2 ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-111/MediaObjects/13661_2012_Article_223_IEq174_HTML.gif. (3.24) and (3.26) are obtained by repeating the above bootstrap argument and the Schauder estimate. This completes the proof of Theorem 3.1. □

Declarations

Acknowledgements

The author thanks the anonymous referee for a careful review and constructive comments. This work was supported by the Qinghai Provincial Natural Science Foundation (2011-Z-915).

Authors’ Affiliations

(1)
Department of Basic Research, Qinghai University

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