数理解析研究所講究録
第 1984 巻 2016 年 9-22

9Asymptotic Behavior of Solutions for Semilinear Volterra
Diﬀusion Equations with Spatial Inhomogeneity
YUSUKE YOSHIDA
Department of Pure and Applied Mathematics
University,
3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, JAPANWasedaE-mail address: ypau-yoshida@suou.waseda.jp

1

Introduction

In this paper we consider the following logistic diﬀusion equation with spatially inhomogeneous coeﬃcients and continuously delay term:(P)$\{\begin{array}{ll}u_{t}=div\{d(x)\nabla u\}+u\{a(x)-b(x)u-c(x)k*u(t)\} in \Omega\cross(0, \infty) ,Bu=0 on \partial\Omega\cross(0, \infty) ,u 0)=u_{0} in \Omega,\end{array}$with suﬃciently smooth boundary
where is a bounded domain inwith $b\geq 0$ and $c\geq 0$ in andfunctions of class$\mathbb{R}^{N}$$\Omega$$\partial\Omega,$

$a,$

$b$

and are$c$$\Omega$$L^{\infty}(\Omega)$$k*u(t) := \int_{0}^{t}k(t-s)u(s)ds.$

with $\alpha\in(0,1)$ . Boundary

A diﬀusion coeﬃcient is a positive function of classoperator represents the following boundary condition$d$$C^{1+\alpha}(\overline{\Omega})$$B$$Bu=u$

where$\partial/\partial n$of class$Bu=\partial u/\partial n+\beta(x)u,$

and is a nonnegative function
denotes the exterior normal derivative to. Moreover, is assumed to be a nonnegative function of class$\partial\Omega$$C^{1+\alpha}(\partial\Omega)$$L^{1}(0, \infty)$or$\beta$$C^{1}(0, \infty)\cap$$k$satisfying$\int_{0}^{\infty}k(t)dt=1$

(1.1)

.

Our problem (P) appears in ecology and denotes the population density of a biological
species. Throughout this paper, we always assume$u$$(A_{i}1)u_{0}$is a nonnegative (not identically zero) function of class$L^{\infty}(\Omega)$,(A.2) $\inf_{\lambda\in 11}\{b(x)+c(x)\}>0.$, then (P) is an initial boundary value problem for a spatially inhomogeneousIflogistic diﬀusion equation. In this case, the dynamics of solutions of (P) is well known
However, it is more realistic to take account of the past(see Cantrell-Cosner [3, 4, 5
information in the study of population biology. The term $k*u(t)$ is sometimes called ahereditary term and describes eﬀects from the past to the present. Naturally, the following
two functions are typical delay kernels in mathematical biology:$c\equiv 0$

$k$

(K.1)

$k(t)^{-}=(1/T)e^{-t/T},$(K.2)$k(t)=(t/T^{2})e^{-t/T}.$10
Here, (K.1) and (K.2) are called a weak delay kernel and a strong delay kernel, re\’{s}pectively.
For instance, they appear in the bacteria model (for details, see Iida [8]).
Our main purpose is to study(P.1) Existence and uniqueness of global solutions of (P),(P.2) Asymptotic behavior of solutions as$tarrow\infty,$(P.3) Existence, uniqueness and stability of positive stationary solutions.

Whenand are constants, $(P.1)-(P.3)$ are studied by many authors and lots ofresults are obtained (see e.g. [8, 10, 14, 17, 18] with homogeneous Neumann boundary
condition and [15, 20] with homogeneous Dirichlet boundary condition). In addition,some systems of Volterra diﬀusion equations have also been studied in [1, 19]. However,there are few results for (P) under inhomogeneous environment. So our main purpose
is to study Volterra diﬀusion equations with spatial inhomogeneity. We have to develop
some devices and tools to discuss the spatial inhomogeneity. Details for our arguments
will be found in the paper of Yoshida-Yamada [21].
The plan of this paper is as follows. In Section 2, we will introduce our results. They
are concerned with $(P.1)-(P.3)$ and main results are Theorems 2.4, 2.5 and 2.7. Consider
the following eigenvalue problem:$d$$a,$ $b,$ $c$(EP)$\{\begin{array}{ll}-div\{d(x)\nabla\psi\}-a(x)\psi=\lambda\psi in \Omega,B\psi=0 on\partial\Omega.\end{array}$Letdenote the principal eigenvalue of (EP). For the proof of some theorems,
the sign of is important. So in Section 3, we will discuss some suﬃcient conditions for. In Sections 4, 5 and 6, we will prove Theorems 2.4, 2.5 and 2.7, respectively.
$\lambda_{1}\equiv\lambda_{1}(a, d)$$\lambda_{1}$$\lambda_{1}<0$

Notation

For $p\in[1, \infty],$$L^{p}(\Omega)$

denotes the Banach space of measurable functions$\Vert u\Vert_{p,\ddagger 1}$$:= \{\int_{tl}|u(x)|^{p}dx\}^{1/p}<\infty$$\Vert u\Vert_{\infty,t)}:=ess\sup_{x\in 1\}}|u(x)|<\infty$

$u$

in

$\Omega$with norm

if $p\in[1, \infty)$ ,if $p=\infty.$

If there is no confusion, then we will omit the subscript . For each $p\in[1, \infty$ ) andinteger $k\in[1, \infty$ ),denotes the usual Sobolev space of measurable functions in. Its normsuch that and its distributional derivatives up to order belong tois defined by$\Omega$$W^{k,p}(\Omega)$$\Omega$

$u$

$k$

$u$

$L^{p}(\Omega)$$\Vert u\Vert_{k,p,t1}=(\sum_{|\alpha|\leq k}\Vert D^{\alpha}u\Vert_{p}^{p})^{1/p}$

where a denotes a multi-index for derivatives. If there is no confusion, then we will also
. Moreover,instead ofomit the subscript . We sometimes write, wheredenotes the space of infinitelyindenotes the closure of$W^{k,2}(\Omega)$$H^{k}(\Omega)$$\Omega$$C_{0}^{\infty}(\Omega)$$W^{k,p}(\Omega)$$C_{0}^{\infty}(\Omega)$$W_{0}^{k,p}(\Omega)$

11
diﬀerentiable functions in with compact support in . In the same way as, wesometimes write
instead of.
Let be any subinterval of) and let $X$ be any Banach space. Denote by $C(I;X)$
the space of $X$ -valued strongly continuous functions in . For any positive integer$C^{j}(I;X)$ denotes the space of functionsof class $C(I;X)$ such that is -times strongly
continuously diﬀerentiable in$\Omega$$W_{0}^{k,2}(\Omega)$$H_{0}^{k}(\Omega)$$I$$H^{k}(\Omega)$$\Omega$$[0, \infty$

$I$

$j,$

$u$

$j$

$u$

$I.$

2

Main results

Let $p>1$ be fixed. Define a closed, linear and elliptic operator$D(A)$ by$A$

with dense domain$Au=-div\{d(x)\nabla u\}$and$D(A)=\{\begin{array}{ll}W_{0}^{1,p}(\Omega)\cap W^{2,p}(\Omega) if Bu=u,\{v\in W^{2,p}(\Omega)|Bu=0 on \partial\Omega\} if Bu=\partial u/\partial n+\beta(x)u.\end{array}$For each $\mu\in[0$ , 1 , we introduce the fractional power spaces
equipped with thegraph norm ofin the standard manner. If $p> \max\{1, N/2\}$ , then$D(A^{\mu})$$]$$A^{\mu}$(2.1)

with $\nu\in[0, 2\mu-(N/p)$ ).

$D(A^{\mu})\subset C^{\nu}(\overline{\Omega})$

For the proof of (2.1), see Henry [6] or Pazy [9]. It is well known that $-A$ generates
an analytic semigroupin. Then we can establish the global exi’stence
theorem.
$L^{p}(\Omega)$$\{e^{-tA}\}_{t\geq 0}$

Theorem 2.1. Let $p> \max\{1, N/2\}$ . Then (P) has a unique solution$u$in the class$u\in C([O, \infty);L^{p}(\Omega))\cap C^{1}((0, \infty);L^{p}(\Omega))\cap C((0, \infty);D(A))$;

which satisfies
$u>0$

in

$\Omega\cross(0, \infty)$

and$\partial u/\partial n<0$in$\overline{\Omega}\cross(0, \infty)$on$\partial\Omega\cross(0, \infty)$

if $Bu=u$ , and
$u>0$

if

$Bu=\partial u/\partial n+\beta(x)u$. Moreover, if $\inf_{x\in\Omega}b(x)>0$ , then$u\leq m$$in$$\Omega\cross(0, \infty)$

,

(2.2)

where $m= \max\{\Vert u_{0}\Vert_{\infty}, \sup_{x\in Jl}\{a(x)/b(x)\}\}.$
This theorem can be proved in the standard manner. For details, see for instance [7]and [18].
By a stationary solution of (P) we mean any solution of(SP)$\{\begin{array}{ll}div\{d(x)\nabla\varphi\}+\varphi[a(x)-\{b(x)+c(x)\}\varphi]=0 in \Omega,B\varphi=0 on \partial\Omega,\end{array}$(note (1.1)). We will look for positive solutions of (SP).

12
Recall (EP). Then
Chapter 2$\lambda_{1}$is given by the following variational characterization (see [5,$\lambda_{1}=\inf_{\psi\in H^{1}(tl) ,\Vert\psi||_{2}=1}\{\int_{tl}d(x)|\nabla\psi|^{2}dx+\int_{\partial t1}d(x)\beta(x)\psi^{2}d\sigma-\int_{ll}a(x)\psi^{2}dx\}$if$B\psi=\partial\psi/\partial n+\beta(x)\psi$, where$\sigma$denotes a surface element, while$\lambda_{1}=\inf_{\psi\in H_{0}^{1}(\Omega) ,\Vert\psi||_{2}=1}\{\int_{l}d(x)|\nabla\psi|^{2}dx-\int_{1l}a(x)\psi^{2}dx\}$

if $B\psi=\psi.$
Then we can obtain the existence and uniqueness of a positive solution of (SP).
, whereis
Theorem 2.2. Problem (SP) has a positive solution if and only if
the principal eigenvalue of (EP). Moreover, when exists, it is uniquely determined andit satisfies$\lambda_{1}<0$$\varphi$$\lambda_{1}$$\varphi$$0<\varphi\leq M$

$in$

(2.3)

$\Omega$and$\{\begin{array}{l}\partial\varphi/\partial n<0 on \partial\Omega if B\varphi=\varphi,0<\varphi\leq M on \partial\Omega if B\varphi=\partial\varphi/\partial n+\beta(x)\varphi,\end{array}$where $M= \sup_{x\in\Omega}\{a(x)/\{b(x)+c(x)\}\}.$
requires $\sup_{x\in t\}}a(x)>0,$ $M$ is a positive number.
Remark 2.1. Since
Theorem 2.2 can be proved as an application of the monotone method (see Sattinger[13, Theorem 2.1]).
We can show the following result on the asymptotic behavior of solutions for (P) in, namely the case of nonexistence of positive stationary solution:
the case of$\lambda_{1}<0$$\lambda_{1}\geq 0$

Theorem 2.3. Assume$\inf_{x\in\Omega}b(x)>0$

Then every solution

$u$

and

$\lambda_{1}\geq 0$

or

$c$

and

$k$

and

$\lambda_{1}>0$

.

(2.4)

of (P) satisfies$\lim_{tarrow\infty}u(t)=0$

Proof. Since$\inf_{x\in\Omega}b(x)=0$

uniformly in 9.

are nonnegative, the positivity of implies$u$$u_{t}\leq div\{d(x)\nabla u\}+u\{a(x)-b(x)u\}.$Consider the following problem:$\{\begin{array}{ll}v_{t}=div\{d(x)\nabla v\}+v\{a(x)-b(x)v\} in \Omega\cross(0, \infty) ,Bv=0 on\partial\Omega\cross(0, \infty) ,v 0)=\Vert u_{0}\Vert_{\infty} in \Omega.\end{array}$Owing to (2.4), the theory of dynamical systems shows$\lim_{tarrow\infty}v(t)=0$

uniformly in 9.

Since the comparison theorem (see e.g. Smoller [16]) shows $u\leq v$ , the conclusion easily
follows.
$\square$13, which assesses that there exists a
In what follows, we will discuss the caseunique positive stationary solution of (SP). First, we will consider the case $\inf_{x\in\Omega}b(x)>$O. Denote by the Laplace transform of :$\lambda_{1}<0$$\varphi$$\hat{k}$$k$$\hat{k}(\lambda)=\int_{0}^{\infty}e^{-\lambda t}k(t)dt.$

Then we can prove the global attractivity of$\varphi$of (SP).

and $tk\in L^{1}(0, \infty)$ . Furthermore, assume
Theorem 2.4. Assume $\inf_{x\in t)}b(x)>0,$
such thatthat there exists a positive constant$\lambda_{1}<0$$k_{0}$$b(x)+{\rm Re}\hat{k}(i\eta)c(x)\geq k_{0}$Then every solution

$u$

for

$x\in\Omega$and$\eta\in \mathbb{R}$

(2.5)

.

of (P) satisfies$\lim_{tarrow\infty}u(t)=\varphi$

uniformly in$\overline{\Omega}$

.

(2.6)

Recall special kernels (K.1) and (K.2). Then both kernels satisfy $tk\in L^{1}(0, \infty)$ .
Moreover, for (K. 1),$\inf_{\eta\in \mathbb{R}}{\rm Re}\hat{k}(i\eta)=\inf_{\eta\in \mathbb{R}}{\rm Re}(\frac{1}{1+i\eta T})$

(2.7)
$=0,$

and for (K.2),$\inf_{\eta\in \mathbb{R}}{\rm Re}\hat{k}(i\eta)=\inf_{\eta\in \mathbb{R}}{\rm Re}(\frac{1}{1+i\eta T})^{2}$(2.8)$=-\underline{1}$8From (2.7) and (2.8),
attractive if$\varphi$is always globally attractive for (K.1), while for (K.2), it is globally$\inf_{x\in tl}\{b(x)-\frac{c(x)}{8}\}>0$

(2.9)

.

When we consider (P) with spatially homogeneous coeﬃcients and homogeneous Neumannboundary condition, it follows from [18] that, if is given by (K.2), then loses its stability
and that the Hopf bifurcation occurs. However, there is no result on the Hopf bifurcationin other cases.
We can also consider (P) for the case $\inf_{x\in Jl}b(x)=0$ . One of the diﬃculties of thiscase is to derive a priori estimate of$k$$\varphi$$u.$

and $tk\in L^{1}(0, \infty)$ . Assume
Theorem 2.5. Let $\inf_{x\in tl}b(x)=0,$. Then every solution of (P) satisfies$\lambda_{1}<0$$\eta\in \mathbb{R}$$u$$\sup_{t\in[0,\infty)}\Vert u(t)\Vert_{\infty}<\infty.$${\rm Re}\hat{k}(i\eta)\geq 0$

for

14
Repeating the proof of Theorem 2.4, we can also obtain the following result:
Theorem 2.6. In addition to the assumptions of Theorem 2.5, assume (2.5). Then every
solution of (P) satisfies$u$$\lim_{tarrow\infty}u(t)=\varphi$

uniformly in$\overline{\Omega}.$

Recall that if is defined by (K.1) (resp. (K.2)), it satisfies (2.7) (resp.
). Thenboth of (K.1) and (K.2) cannot satisfy (2.5). This implies that Theorem 2.6 is inconvenientfrom the viewpoint of the application. By putting additional assumptions, we can improve
Theorem 2.6 as follows.
$k$$(2.8)_{t}$Theorem 2.7. In addition to the assumptions of Theorem 2.5, assume $k(O)\neq 0$ and$dk/dt)\in L^{1}(0, \infty)$ . Furthermore, assume that there exist positive
constants andsuch that $c(x)\geq c_{0}$ forand$k$$c_{0}$$x\in\Omega$$k_{1}$

for${\rm Re}\{\hat{k}(i\eta)\}^{-1}\geq k_{1}$Then every solution$u$Suﬃcient conditions foruniformly in$\overline{\Omega}.$$\lambda_{1}<0$

In this section, we will search some suﬀcient conditions for$\Omega_{0}=\{x\in\Omega|a(x)>0\}$and always assume(2.10).

of (P) satisfies$\lim_{tarrow\infty}e\}(t)=\varphi$

3$\eta\in \mathbb{R}$$\Omega_{0}\neq\emptyset$$\lambda_{1}<0$

. Set

,

(3.1)

in this section. Consider the following eigenvalue problem:$\{\begin{array}{l}-div\{d(x)\nabla\rho\}=\mu a(x)p in \Omega,(3.2)B\rho=0 on \partial\Omega.\end{array}$Letdenote the positive principal eigenvalue of (3.2). It is given by thefollowing variational characterization (see e.g. [5]):$\mu_{1}^{+}\equiv\mu_{1}^{+}(a, d)$$\frac{1}{\mu_{1}^{+}}=\rho\in H^{1}(tl)\sup_{\rho\neq 0}\frac{\int_{Jl}a(x)\rho^{2}dx}{\int_{tl}d(x)|\nabla p|^{2}dx+\int_{\partial t)}d(x)\beta(x)p^{2}d\sigma}$

if $B\psi=\partial\psi/\partial n+\beta(x)\psi$ , while(3.3)$\frac{1}{\mu_{1}^{+}}=\rho\in H_{0}^{1}(\Omega)\sup_{\rho\neq 0}\frac{\int_{\Omega}a(x)\rho^{2}dx}{\int_{tl}d(x)|\nabla\rho|^{2}dx}$

if $B\psi=\psi$ . Note that if$Bp=\partial p/\partial n$, then$\mu_{1}^{+}$

exists if and only if(3.4)$\int_{l}a(x)dx<0$ .

For the proof of (3.4), see, e.g., [5]. The relation between
following proposition (see [5, Theorem 2.6]):$\lambda_{1}$and$\mu_{1}^{+}$

is given by the

15be the principal eigenvalue of (EP) and let
Proposition 3.1. Let(3.2).
principal eigenvalue ofbe the positive$\mu_{1}^{+}$$\lambda_{1}$(i) If $B\psi=\psi$ or(ii) If$B\psi=\partial\psi/\partial n+\beta(x)\psi(\beta\not\equiv 0)$$B\psi=\partial\psi/\partial n$with (3.4), then$\lambda_{1}<0$, thenif and only if$\int_{\Omega}a(x)dx>0$

then$\lambda_{1}<.0$

if and only if. If$\mu_{1}^{+}<1.$$B\psi=\partial\psi/\partial n$$\mu_{1}^{+}<1$

with
(3.5)

,

$\lambda_{1}<0.$Then the following result is obtained:

Proposition 3.2. Define$\Omega_{0}$$B\psi=\partial\psi/\partial n+\beta(x)\psi(\beta\not\equiv 0)$by (3.1). Let $B\psi=\psi$ or. Then there exists a positive constantfor any . satisfying
Proof. We will only discuss the case$d$with (3.4) orsuch that$B\psi=\partial\psi/\partial n$$d^{*}$$\lambda_{1}<0$$\Vert d\Vert_{\infty,1l_{0}}<d^{*}$$B\psi=\psi$. The other cases can be handled similarly.

. Take any function
Take any connected set
the natural extension of . Observing (3.3), we can estimate$\rho\in H_{0}^{1}(\Omega_{0}^{*})$$\Omega_{0}^{*}\subset\Omega_{0}$and let$\tilde{\rho}:\Omega_{0}^{*}arrow \mathbb{R}$be$\rho$$\frac{1}{\mu_{1}^{+}}\geq\sup_{\rho\neq 0}\frac{\int_{ll}a(x)\tilde{p}^{2}dx}{\int_{\zeta)}d(x)|\nabla\tilde{\rho}|^{2}dx}\rho\in H_{O}^{1}(l_{O}^{*})$$\int_{\zeta)_{0}^{*}}a(x)\rho^{2}dx$$\geq\Vert d\Vert_{\infty,\Omega_{0_{\rho\in H_{0}^{1}(t)_{0}^{*})}}^{*\sup_{\rho\neq 0}}}^{-1}\int_{tl_{0}^{*}}|\nabla\rho|^{2}dx$

Choose$d^{*}$as$d^{*}= \rho\in H_{O}^{1}(Jl_{0}^{*})\sup_{\rho\neq 0}\frac{\int_{\Omega_{0}^{*}}a(x)\rho^{2}dx}{\int_{(t_{0}^{*}}|\nabla\rho|^{2}dx}.$Thenconclusion.

$\mu_{1}^{+}<1$

for any$d$satisfying$\Vert d\Vert_{\infty,\Omega_{0}^{*}}<d^{*}$. Therefore, Proposition 3.1 yields the$\square$Propositions 3.1 and 3.2 imply that a positive stationary solution exists if a diﬀusion
is suﬃciently small or (3.5) is achieved with homocoeﬃcients in a favorable habitat
geneous Neumann boundary condition. In ecology, this fact asserts that there is a chance
for a species to survive if the species stays in a favorable habitat$\Omega_{0}$$\Omega_{0}.$164Proof of Theorem 2.4

For the proof of Theorem 2.4, we will follow the argument used by Yamada [20]. Let
be a positive solution of (SP). We introduce the following nonnegative functional:$\varphi$$E(u)= \int_{\zeta)}\varphi^{2}(x)g(u(x)/\varphi(x))dx$(4.1)$= \int_{\zeta\}}\varphi(x)\{u(x)-\varphi(x)-\varphi(x)\log\frac{u(x)}{\varphi(x)}\}dx,$

where$g(u)=u-1-\log u$ .

(4.2)

This functional has also been used in [1].

Lemma 4.1 (cf. [20, Lemma 3.1]). Define $E(u)$ by (4.1). Then any solution$u$satisfiesof (P)$\frac{d}{dt}E(u(t))=-\int_{Il}d(x)\varphi^{2}|\nabla\{\log\frac{u(t)}{\varphi}\}|^{2}dx-\int_{\zeta\}}b(x)\varphi\{u(t)-\varphi\}^{2}dx$$- \int_{t1}c(x)\varphi\{u(t)-\varphi\}k*(u-\varphi)(t)dx$(4.3)$+ \int_{t}^{\infty}k(s)ds\int_{l}c(x)\varphi^{2}\{u(t)-\varphi\}dx.$

Proof. We will only prove the case $Bu=u$ . The other cases can be proved similarly.
Diﬀerentiation of (4.1) with respect to yields$t$$\frac{d}{dt}E(u(t))=\int_{tl}\{1-\frac{\varphi}{u(t)}\}u_{t}(t)\varphi dx$$= \int_{tl}\varphi\{1-\frac{\varphi}{u(t)}\}$

div{d(x)Vu(t)}dx$+ \int_{Jl}\varphi\{u(t)-\varphi\}\{a(x)-b(x)u(t)-c(x)k*u(t)\}dx.$

In view of (1.1),$a(x)-b(x)u-c(x)k*u(t)=a(x)-\{b(x)+c(x)\}\varphi-b(x)(u-\varphi)$$-c(x)k*(u-\varphi)(t)+c(x)\varphi l^{\infty}k(s)ds.$

Since$\varphi$

is a solution of (SP), it follows that$\frac{d}{dt}E(u(t))=\int_{\Omega}\varphi\{1-\frac{\varphi}{u(t)}\}div\{d(x)\nabla u(t)\}dx-\int_{\Omega}\{u(t)-\varphi\}div\{d(x)\nabla\varphi\}dx$$- \int_{Il}b(x)\varphi\{u(t)-\varphi\}^{2}dx-\int_{1}c(x)\varphi\{u(t)-\varphi\}k*(u-\varphi)(t)dx$$+ \int_{t}^{\infty}k(s)ds\int_{\zeta)}c(x)\varphi^{2}\{u(t)-\varphi\}dx.$

17Moreover, we also obtain from the integration by parts$\int_{\zeta\}}\varphi\{1-\frac{\varphi}{u(t)}\}div\{d(x)\nabla u(t)\}dx-\int_{\zeta\}}\{u(t)-\varphi\}div\{d(x)\nabla\varphi\}dx$$=- \int_{\zeta)}d(x)\{|\nabla\varphi|^{2}-\frac{2\varphi}{u(t)}\nabla\varphi\cdot\nabla u(t)-\frac{\varphi^{2}}{u^{2}(t)}|\nabla u(t)|^{2}\}dx$$=- \int_{t1}d(x)\varphi^{2}|\nabla\{\log\frac{u(t)}{\varphi}\}|^{2}dx.$

Therefore, (4.3) follows.

$\square$

Then we are ready to follow the argument in [20]. We will also prepare some regularity
results.

Lemma 4.2. Let be a bounded solution of (P) and let be any positive number. Then
there exist positive constants, independent of , such that for $p>1,$
andand $\mu\in[0$ , 1),$\delta$$u$$K_{1},$$K_{2}$$K_{3}$$t$$t\in[\delta, \infty)$$\Vert A^{\mu}u(t)\Vert_{p}\leq K_{1},$and for $h>0,$(4.4)$\Vert A^{\mu}\{u(t+h)-u(t)\}\Vert_{p}\leq K_{2}h^{\theta}+K_{3}h^{1-\mu}$with $\theta\in(0,1-\mu)$ .
For the proof of Lemma 4.2, see [18, Theorem 3.1] and Rothe [12, Lemma 21].

Proof of Theorem 2.4. We may assume $u_{0}>0$ . Indeed, we can retake $u_{0}=u(T)>0$ for$T>0$ and prove this theorem with slight modification. Integrating (4.3) over [O,T] with
arbitrary number $T>0$ , we have$E(u(T))+ \int_{0}^{T}\int_{t1}d(x)\varphi^{2}|\nabla\{\log\frac{u(t)}{\varphi}\}|^{2}dxdt+\int_{0}^{T}\int_{\zeta)}b(x)\varphi\{u(t)-\varphi\}^{2}dxdt$(4.5)$+ \int_{0}^{T}\int_{1}c(x)\varphi\{u(t)-\varphi\}k*(u-\varphi)(t)dxdt$$=E(u_{0})+ \int_{0}^{T}\int_{t}^{\infty}k(s)ds\int_{l}c(x)\varphi^{2}\{u(t)-\varphi\}dxdt.$

By virtue of (2.2) and (2.3), the second term in the right hand side of (4.5) is estimatedas$\int_{0}^{T}\int_{t}^{\infty}k(s)ds\int_{tl}c(x)\varphi^{2}\{u(t)-\varphi\}dxdt\leq\Vert c\Vert_{\infty}M^{2}(m+M)|\Omega|\int_{0}^{\infty}sk(\mathcal{S})ds.$

Since $tk\in L^{1}(0, \infty)$ , it follows from (4.5) and the above inequality that$\int_{0}^{T}\int_{1l}b(x)\varphi\{u(t)-\varphi\}^{2}dxdt+\int_{0}^{T}\int_{\zeta)}c(x)\varphi\{u(t)-\varphi\}k*(u-\varphi)(t)dxdt\leq K_{4}$

,

(4.6)

18
where
$K_{4}=E(u_{0})+ \Vert c\Vert_{\infty}M^{2}(m+M)|\Omega|\int_{0}^{\infty}tk(t)dt.$

For

:

$v$

$[0, T]arrow \mathbb{R}$, define$v_{T}$

by$v_{T}(t)=\{\begin{array}{ll}v(t) if t\in[O, T],0 if t\in(-\infty, \infty)\backslash [0, T],\end{array}$and for $k:[0, \infty$ )$arrow \mathbb{R}$, define$\tilde{k}$by$\tilde{k}(t)=\{\begin{array}{ll}k(t) if t\in[0, \infty) ,0 if t\in(-\infty, 0) .\end{array}$

Then we can derive the following relation (cf. [18, Lemma 2.2]):$\mathcal{F}(\tilde{k}*\prime)\tau)(\eta)=\hat{k}(i\eta)\mathcal{F}v_{T}(\eta)$

where$\mathcal{F}v$

(4.7)

,

denotes the Fourier transform of :$v$$\mathcal{F}v(\eta)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}e^{-i\eta t}\tau)(t)dt.$

Therefore, making use of (2.5), (4.7), Fubini’s theorem and Parseval-Plancherel’s equality,
we can obtain$\int_{0}^{T}\int_{\Omega}b(x)\varphi\{u(t)-\varphi\}^{2}dxdt+\int_{0}^{T}\int_{\Omega}c(x)\varphi\{u(t)-\varphi\}k*(u-\varphi)(t)dxdt$(4.8)$\geq k_{0}\int_{0}^{T}\int_{\zeta)}|u(t)-\varphi|^{2}\varphi dxdt_{1}$(for details, see [18] and [20]).
Since is arbitrary andis independent of$T$$K_{4}$$T$, (4.6) and (4.8) yield$\varphi^{1/2}(u-\varphi)\in L^{1}((0, \infty);L^{2}(\Omega))$

(4.9)

.

On the other hand, (4.4) shows thatis uniformly continuous inrespect tonorm. The fact, together with (4.9) implies$\varphi^{1/2}(u-\varphi)$$(0, \infty)$with$L^{2}(\Omega)$$\lim_{tarrow\infty}\int_{\Omega}|u(t)-\varphi|^{2}\varphi dx=0$.

(4.10)

Then we can prove (2.6) from (4.10). Its proof is exactly the same as in [20] withreplaced by $A+1.$

Remark 4.1. Take $p>N$ and $\mu\in((p+N)/(2p), 1)$ . Then (2.1) implies$\lim_{tarrow\infty}u(t)=\varphi$

in$C^{1}$(St).

$A$$\square$19
5

Proof of Theorem 2.5

We will prove this theorem along the arguments used in the work of Yamada [20, Proposi, there exists an unique positive stationary
tion 3.3]. We can assume $u_{0}>0$ . Since
solution of (SP). Then integrating (4.3) over $[0, T]$ with $T>0$ , we see$\lambda_{1}<0$$\varphi$(5.1)$E(u(T)) \leq E(u_{0})+\Vert c\Vert_{\infty}M\int_{0}^{T}\int_{t}^{\infty}k(\mathcal{S})ds\int_{)}\varphi u(t)dxdt$

is a convex function (see (4.2)), it is possible toapply Jensen’s inequality (see e.g. [2, pp. 120]) to $E(u(T))$ to get

as in the proof of Theorem 2.4. Since$g$$g( \Vert\varphi\Vert_{2}^{-2}\int_{\zeta)}\varphi u(T)dx)\leq\Vert\varphi\Vert_{2}^{-2}\int_{tl}\varphi^{2}g(\frac{u(T)}{\varphi})dx$(5.2)$=\Vert\varphi\Vert_{2}^{-2}E(u(T))$Put$V(t):= \Vert\varphi\Vert_{2}^{-2}\int_{\Omega}\varphi u(t)dx$.

. Then we obtain from (5.1) and (5.2)$\Vert\varphi\Vert_{2}^{2}g(V(T))\leq E(v_{0})+\Vert c\Vert_{\infty}\cdot\Vert\varphi\Vert_{2}^{2}M\int_{0}^{T}\int_{t}^{\infty}k(s)dsV(t)dt$

By using the idea in [20], it can be shown that for suﬃciently large

(5.3)

.

$T_{0},$$K_{6}:= \Vert c\Vert_{\infty}M\int_{T_{0}}^{\infty}l^{\infty}k(s)dsdt<1.$

Then (5.3) implies that for every $T\geq T_{0},$$g(V(T)) \leq\Vert\varphi\Vert_{2}^{-2}E(u_{0})+\Vert c\Vert_{\infty}M\int_{0}^{T_{0}}\int_{t}^{\infty}k(s)d_{\mathcal{S}}V(t)dt+K_{6}\sup_{t\geq T_{0}}V(t)$Recall that$g$is given by (4.2). Since $K_{6}<1$ , it follows from the above inequality that
(5.4)$\sup_{t\geq T_{0}}V(t)\leq K_{7}$with some$K_{7}$.

. Then it follows from (5.4) that

(5.5)$\sup_{t\geq 0}\int_{\Omega}\varphi u(t)dx\leq K_{8}$with some
Let $r\in(O, 1/2)$ . Then we can obtain from (5.5) that
$K_{8}.$$\sup_{t\geq 0}f_{l}u^{r}(t)dx\leq K_{9}$

(5.6)

,

is a suitable positive constant, independent of (for details, see [20]). Therefore,
where
the result of Rothe [11, Proposition 2] implies the conclusion.
$K_{9}$$t$$\square$20
6

Proof of Theorem 2.7

We will prove this theorem by the same idea as in [20, Theorem 3.5]. We may assume$u_{0}>0$ . By
, there exists a unique positive stationary solution of (SP). Define a
new function as$\lambda_{1}<0$$\varphi$$v$$v(x, t)=k*(u-\varphi)(x, t)$ .

Similarly to the proof of Theorem 2.4, integrate (4.3) over $[0, T]$ with an arbitrary $T>0$ ;then there exists a positive constant, independent of , such that$T$

$K_{10}$$E(u(T))+ \int_{0}^{T}\int_{t)}d(x)\varphi^{2}|\nabla\{\log\frac{u(t)}{\varphi}\}|^{2}dxdt$(6.1)$+ \int_{0}^{T}\int_{tl}b(x)\varphi\{u(t)-\backslash \prime\rho\}^{2}dxdt+\int_{0}^{T}\int_{tl}c(x)\varphi\{u(t)-\varphi\}v(t)dxdt\leq K_{10}.$

Since

$v$

satisfies$v_{t}(t)=k(0)\{u(t)-\varphi\}+k’*(u-\varphi)(t)$ ,(6.2)

it follows that$\frac{1}{2}\frac{d}{dt}.l_{l}c(x)\varphi v^{2}(t)dx=\int_{l}c(x)\varphi v(t)v_{t}(t)dx$$= \int_{11}c(x)\varphi v(t)[k(0)\{u(t)-\varphi\}+k’*(u-\varphi)(t)]dx.$

Integrate this identity over $[0, T]$ :$\frac{1}{2}\int_{0}^{T}\frac{d}{dt}\int_{\zeta\}}c(x)\varphi v^{2}(t)dxdt$(6.3)$= \int_{0}^{T}\int_{\zeta\}}c(x)\varphi v(t)[k(O)\{u(t)-\varphi\}+k’*(u-\varphi)(t)]dxdt.$

In view of $v(O)=0,$$\int_{0}^{T}\frac{d}{dt}\int_{1}c(x)\varphi v^{2}(t)dxdt=f_{l}c(x)\varphi v^{2}(T)dx$$\geq 0.$

Therefore, we can see from (6.3)$- \int_{0}^{T}\int_{\Omega}c(x)\varphi v(t)k’*(u-\varphi)(t)dxdt\leq k(0)\int_{0}^{T}\int_{\Omega}c(x)\varphi\{u(t)-\varphi\}v(t)dxdt$(6.4)$\leq k(0)K_{10},$where we have used (6.1).
Note$\hat{k}’(i\eta)=\int_{0}^{\infty}e^{-i\eta t}k’(t)dt$$=-k(0)+i\eta\hat{k}(i\eta)$

21
and

$\mathcal{F}(\prime J_{T)(\eta)}=\hat{k}(i\eta)\mathcal{F}((u-\varphi)_{T})(\eta)$. Then$\mathcal{F}(\tilde{k}’*(u-\varphi)_{T})(\eta)=\{-k(0)+i\eta\hat{k}(i\eta)\}\{\hat{k}(i\eta)\}^{-1}\mathcal{F}(\uparrow)\tau)(\eta)$

.

(6.5)

By virtue of $(6,5)$ , Fubini’s theorem and Parseval-Plancherel’s equality, similarly in the
proof of Theorem 2.4, we can show$- \int_{0}^{T}\int_{tl}c(x)\varphi v(t)k’*(u-\varphi)(t)dxdt$$=- \int_{\zeta f}c(x)\varphi\int_{-\infty}^{\infty}v_{T}(t)\tilde{k}’*(u-\varphi)_{T}(t)dtdx$$=- \int_{tl}c(x)\varphi\int_{-\infty}^{\infty}{\rm Re} \mathcal{F}(v_{T})(\eta)\mathcal{F}(\tilde{k}’*(u-\varphi)_{T})(\eta)d\eta dx$$=k(0) \int_{\Omega}c(x)\varphi\int_{-\infty}^{\infty}{\rm Re}\{\hat{k}(i\eta)\}^{-1}|\mathcal{F}(v_{T})(\eta)|^{2}d\eta dx.$

Therefore, (6.4) implies$c_{0}k_{1} \int_{0}^{T}\int_{)}\varphi v^{2}(t)dxdt\leq K_{10},$(note (2.10)). Since$T$

is arbitrary and$K_{10}$is independent of$\varphi^{1/2}v\in L^{1}((0, \infty);L^{2}(\Omega))$$T$, this fact implies

(6.6)

.

One can prove the uniformly continuity of $v(t)$ with respect toHence, it follows from (6.6) that$\lim_{tarrow\infty}\varphi^{1/2}v(t)=0$

In the same manner as [20] (replace

$A$

in

$L^{2}(\Omega)$$t$from (6.2) (see [20]).

.

by $A+1$ ),$\lim_{tarrow\infty}v(t)=0$

uniformly in 9.

(6.7)

The rest of the proof is essentially the same as Yamada [20]. So we omit the details.

$\square$

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