Quasilinear reaction diffusion systems with mass dissipation

1.   Introduction

Quasilinear reaction diffusion system is given by

for $1\leq i\leq N$, where $\tau = (\tau_i)\in \mathbb{R}_+^N$, $u = (u_1(x, t), \ldots, u_N(x, t))\in \mathbb{R}^N$, $\Omega\subset{\bf R}^n$ is a bounded domain with smooth boundary $\partial\Omega$, $\nu$ is the outer unit normal vector, and $u_0 = (u^0_{i}(x))\not\equiv 0$ is the initial value sufficiently smooth. For the nonlinearity it is assumed that

is smooth, and $f = (f_i(u)): \overline{\mathbb{R}}_+^N\rightarrow \mathbb{R}^N$ is locally Lipschitz continuous and quasi-positive:

We have, therefore, unique existence of a positive classical solution local in time. Our purpose is to extend this solution global in time. This question is posed in ^{[12,14,17,33,37]} with a positive result.

Main assumption below is the total mass dissipation

which implies

In the semilinear case when $d_i(u) = d_i > 0$ for $1\leq i\leq N$, if $f = (f_i(u))$ is of quadratic growth rate;

then $u = (u_i(x, t))\geq 0$ is uniformly bounded and hence global in time,

This result is a direct consequence of (1.5) for $n = 1$ (^[10]), and the cases $n = 2$ and $n\geq 3$ are proven by ^[28,36] and ^[6,7], respectively. For the quasi-linear case of (1.1), however, several tools of the latter approach require non-trivial modifications ^[20], such as regularity interpolation ^[13] or Souplet's trick ^[31]. Here we examine the validity of the former approach.

So far, global in time existence of the weak solution has been discussed in details. In ^{[2,3,5,8,9,29]} it is observed that by an appropriate logarithmic change of variables (1.1) can be transformed into a system with a symmetric and positive definite diffusion matrix. In ^[3], furthermore, it is shown that

where

and $\mathcal{D}(t)$ stands for the energy dissipation, which induces $u_i\log u_i\in L^ \infty (0, T;L^1(\Omega))$ and $\nabla\sqrt{u_i}\in L^2(\Omega_T)$. This structure is used in ^[4,12], to derive existence of the weak solution global in time to (1.1) for an arbitrary number of competing population species,

with non-negative and positive constants $a_{ij}$ for $1\leq i, j \leq N$ and $a_{ij}$ for $1\leq j\leq N$, respectively, under the detailed balance condition

for positive constants $\pi_i$, $1\leq i\leq N$.

The fundamental assumption used in this approach is

for

where $c_0$, $\delta$, $C$ are positive constants. This assumption induces a uniform estimate of the solution in $L\log L$ norm.

Theorem 1. Let $d = (d_i(u))$ satisfy (1.2) and (1.9)–(1.10). Assume that $d(u)\cdot u$ is bounded above and below by positive constants $\delta$, $C$,

Let, furthermore, $f = (f_i(u))$ satisfy (1.3)–(1.4) and be of quadratic growth rate in the sense that it satisfies (1.6) and

Then, it holds that

for $u = (u_i(\cdot, t))$.

Here we use the fact that (1.13) means

by $u_i\geq 0$ and (1.15), see ^[11].

Theorem 2. Let $d = (d_i(u))$ satisfy (1.2) and (1.9)–(1.10). Assume that it is of linear growth rate,

with $\delta > 0$. Assume, furthermore, the cubic growth rate of $f = (f_i(u))$:

Then (1.13) holds.

Under the setting of the above Theorem, the classical solution exists locally in time if the initial value is sufficiently regular. It there is a uniform estimate of the solution:

this classical solution extends after $t = T$, see ^[16].

At this stage, the method of ^[28,36] ensures $L^q$ estimate of the classical solution under the cost of low space dimension. We require, however, an additional assumption to execute Moser's iteration ^[1].

Letting

we obtain

and therefore, (1.1) is reduced to

The diffusion matrix $A = (A_{ij}(u))$ is not necesarily symmetric nor positive definite. Our assumption is

for $A_\alpha(u) = (A_{ij}^\alpha(u))$ and $A_{ij}^\alpha(u) = A_{ij}(u)(u_i/u_j)^{\alpha}$, where $I$ denotes the unit matrix and $\delta$ is a positive constant.

Theorem 3. If $f = (f_i(u))$ is of quadratic growth satisfying (1.6) and (1.12). Suppose (1.13) for the solution. Then, (1.18) implies

for any $1\leq q < \infty$.

The Shigesada-Kawasaki-Teramoto (SKT) model ^[18,30] describes separation of existence areas of competing species. There, it is assumed that $N = 2$,

and

where $a_{ij}$, $a_i$, $b_i$, $c_i$ are non-negative constants for $i, j = 1, 2$ and $a_{10}$, $a_{20}$ are positive constants.

Equalities (1.20) in SKT model are due to cross diffusion where the transient probability of particle is subject to the state of the target point ^[26,35], while equalities (1.21) are Lotka-Volterra terms describing competition of two species in the case of

The Lotka-Volterra reaction-diffusion model without cross diffusion is the semilinear case, where $d_i(u) = d_i$, $i = 1, 2$, are positive constants as $a_{ij} = b_{ij} = 0$ in (1.20). For this system, any stable stationary solution is spatially homogeneous if $\Omega$ is convex ^[15], while there is (non-convex) $\Omega$ which admits spatially inhomogeneous stable stationary solution ^[24]. Coming back to the SKT model, we have several results for structure of stationary solutions to a shadow system ^{[21,22,23,25]}. There is also existence of the solution to the nonstationary SKT model global in time and bounded in $H^2$ norm if

(^[38]). Obivously, Theorems 1 and 2 are not applicable to this system without total mass dissipation (1.4). Such $f = (f_i(u))$, admitting linear growth term in (1.4), is called quasi-mass dissipative. Global in time existence of the solution without uniform boundedness is the question for the general case of quasi-mass dissipation.

We have the following theorem valid to such reaction under

Theorem 4. Let $d = (d_i(u))$ satisfy (1.24), and assume (1.3) and

for $f = (f_i(u))$. Then, it holds that $T = +\infty$ for any space dimension $n$.

Concluding this section, we examine the condition posed in above theorems, for $d = (d_i(u))$ given by (1.20). First, for (1.9)–(1.10), we confirm

Then (1.10) reads

or equivalently,

Inequality (1.26) means

and therefore,

is the condition of (1.20) for (1.9)–(1.10).

For (1.18), second, we note

to confirm

for $A^0_\alpha(u) = \mbox{diag}(a_{10}u_1, a_{20}u_2)$ and

Hence (1.18) follows from $A_\alpha^1+{}^tA_\alpha^1\geq 0$, or

which is reduced to

This condition is thus satisfied if

Finally, condition (1.14) holds if

From (1.27), particularly (1.29), cross diffusion is essentially excluded in the application of Theorems 2, 3, 4 to (1.20).

2.   Proof of Theorems

We begin with the following proof.

Proof of Theorem 4. By (1.17) we obtain

for $p > 0$ and $1\leq i\leq N$, and therefore,

by (1.25) and we remind that $u = (u_i)\geq 0$. Since

it holds that

By (1.24) we have

which implies

Then we obtain

by making $p\uparrow +\infty$ with $C_3 = C_3(\Vert u_0\Vert_\infty)$, and hence $T = +\infty$.

Three lemmas are needed for the proof of the other theorems.

Lemma 5. Assume (1.3). Then inequality (1.12) implies

The second inequality of (1.15), similarly, implies

Proof. The former part is proven in ^[36]. The latter part follows similarly, which we confirm for completeness. In fact, given $u = (u_i)\geq 0$, put

It holds that

by (1.3), where

We assume $\vert u\vert \geq 1$ because inequality (2.3) is obvious for the other case of $\vert u\vert\leq 1$. It may be also assumed that $0 < s_i\leq 1$ for $u_i = s_i\vert u\vert$. Then we obtain

by $\vert u\vert\geq 1$, (1.4), and (2.5). It thus holds that (2.4) for $\vert u\vert \geq 1$ as

by $0 < s_i\leq 1$, $1\leq i\leq N$.

Lemma 6. If $d = (d_i(u))$ satisfies (1.2), (1.4), and (1.14), then it holds that

If $d = (d_i(u))$ satisfies (1.2), (1.4), and (1.11), it holds that

Proof. The latter part is well-known ^[27,34]. The former part follows similarly, which we again confirm for completeness. In fact, (1.4) implies

and hence

where $(\cdot, \cdot)$ denotes the inner product in $L^2(\Omega)$. Then it follows that

and hence the result.

The following lemma has been used for construction of weak solution global in time ^[4,12].

Lemma 7. Under the assumption of (1.9)–(1.10) it holds that

Proof. Let

be the Onsager matrix, where $A = (A_{ij}(u))$ and $H(u) = diag\, (u_1^{-1}, \ldots, u_N^{-1})$. Regard $B = B(w)$ for

and observe that (1.9)–(1.10) implies

by (1.16). We obtain, furthermore,

for $1\leq i\leq N$ by (1.17).

Put

Then we obtain

by (2.10), and hence (2.8).

Proof of Theorems 1 and 2. These theorems are a direct consequence of Lemmas 5, 6, and 7.

Proof of Theorem 3. Any $\varepsilon > 0$ admits $C_\varepsilon$ such that

See Chapter 4 of ^[32] for the proof. We have, on the other hand,

by (1.6), (1.18), (2.1), and (2.2), where

Letting

we obtain

with $c_3 > 0$. Apply the Gagliardo-Nirenberg inequality for $n = 2$,

Here we notice Wirtinger's inequality to deduce

In this inequality $C$ on the right-hand side is independent of $1\leq p < \infty$, beucase it then follows that $2 < r\leq 3$.

For $p = 1$ we use (2.16) to derive

for any $\varepsilon > 0$ by (2.12). Then it follows that

For $p > 1$, second, there arises $\frac{p+3}{p+1} < 1$, and hence (2.13) and (2.16) implies

By (2.17)–(2.18) it holds that (1.19) for any $1\leq q < \infty$.

Remark 1. For system of chemotaxis in two space dimension, inequality (1.19) for $q = 3$ implies uniform boundedness of the chemical term by the elliptic regulariy, which replaces the right-hand side on (2.13) by a constant times $1+\Vert u\Vert_{p+1}^{p+1}$. Then Moser's iteration scheme induces (1.19) for $q = \infty$. See Chapter 11 of ^[32] for details. For the case of constant $d_i$ in (1.1), on the other hand, the semigroup estimate is applicable as in ^[19]. If $n = 2$, for example, inequality (1.19) for $q = 2$ implies that for $q = \infty$. Such parabolic estimate to (1.1) will be discussed in future.

Acknowledgements

The first author is supported by the Austrian Science Fund (FWF): F73 SFB LIPID HYDROLYSIS. The second author is supported by JSPS core-to-core research project, Kakenhi 16H06576, and Kakenhi 19H01799.

Conflict of interest

The authors declare no conflict of interest.