In the past decades, many physical phenomena have been formulated into nonlocal mathematical models and studied by many authors. In particular, there exist many articles dealing with properties of solutions to semilinear or degenerate parabolic equations with homogeneous Dirichlet boundary condition (see [1-4] and references therein). For example, Li et al. [5] studied the nonlinear nonlocal porous medium equation
which is subjected to homogeneous Dirichlet boundary condition. Under appropriate hypotheses for initial data, the following results were obtained:
(1) In case of $p+q < m$ or $p+q=m$ and $a$ is sufficiently small, then there exists a global positive classical solution.
(2) In case of $p+q>m$, then the solution either blows up in finite time if initial value $u_0$ is large enough or exists globally provided that $u_0$ is sufficiently small.
However, there are some important phenomena formulated into parabolic equations with nonlocal boundary conditions in mathematical modeling such as thermoelasticity theory (see [6]). In 1986, Friedman [7] studied the following parabolic equation with nonlocal boundary condition:
He gave the global existence of solutions provided that $\displaystyle\int_\Omega | g(x, y)| \mathrm{d}y < 1$ for $x\in\partial\Omega$ and $c(x)\leq 0$. Recently, Gladkov and Kim [8] extended the type of (1.2), they considered the problem of the form:
with $p, l>0$. And some criteria for the existence of global solution as well as for the solution to blow-up in finite time were obtained. For other works on parabolic equations with nonlocal nonlinear boundary conditions, we refer readers to [9-11] and references therein.
Motivated by those of works above, we will extend the works in [5] and investigate the positive solution of the following degenerate nonlinear nonlocal porous medium equation:
where $\Omega \subset \mathbb{R}^N$ is a bounded domain with sufficiently smooth boundary $\partial\Omega$, $a>0$, $m>1$, $p, q \geq 0$, $l>0$. The aim of this paper is to understand how the weight function $g(x, y)$ and the nonlinear term $u^l(y, t)$ in the boundary condition play substantial roles in determining blow-up or not of solutions and give some criteria for the existence of global solutions as well as for a blow-up of solutions in a finite time.
In this paper, we give the following hypotheses (H1)-(H4):
(H1) $u_0(x)\displaystyle \in C^{2+\alpha}(\Omega)\cap C(\overline{\Omega}), \alpha \displaystyle\in (0, 1), u_0(x)>0$ in $\Omega$, $u_0(x)=\displaystyle\int_\Omega g(x, y) u_0^l(y)\mathrm{d}y$ on $\partial\Omega$.
(H2) $g(x, y)$ is a nonnegative and continuous function defined for $x\in \partial\Omega, y\in \Omega$.
(H3) $\Delta u_0^m + a u_0^p \displaystyle\int_\Omega u_0^q \mathrm{d}x>0, x\in\Omega$.
(H4) There exists a constant $\rho \geq \rho_0$ such that
where
The plan of this paper is as follows: In Section 2, we establish the comparison principle and the local existence. Some criteria regarding to global existence and finite time blow-up for problem (1.4) are given in Section 3. In Section 4, we give the blow-up rate estimate for the special case of $l=1$.
At first, we give the definition of sub-solution and super-solution of (1.4) and comparison principle. Let $Q_T=\Omega \times (0, T), S_T =\partial\Omega \times (0, T)$, $\overline{Q}_T=\overline{\Omega}\times [0, T)$.
Definition 1 A function $\tilde{u}(x, t)\in C^{2, 1}(Q_T)\cap C(\overline{Q}_T)$ is called a sub-solution of $(1.4)$ on $Q_T$ if
Similarly, a super-solution $\bar{u}(x, t)$ of (1.4) is defined by the opposite inequalities. We say that $u$ is a solution of (1.4) in $Q_T$ if both a sub-solution and a super-solution of (1.4) in $Q_T$. Further, we say that $u$ is a global solution of (1.4) if it is a solution of (1.4) in $Q_T$ for any $T>0$. By similar arguments in [12] we have
Lemma 1 Suppose that $w(x, t)\in C^{2, 1}(Q_T)\cap C(\overline{Q}_T)$ and satisfies
where $d(x, t), c_i(x, t)~ (i=1, 2, 3, 4)$ are bounded functions and $d(x, t), c_i(x, t)\geq 0~ (i=1, 2, 3, 4)$ in $Q_T$, $c_5(x, y)\geq 0$ for $x\in\partial\Omega, y\in\Omega$ and is not identically zero. Then, $w(x, 0)>0$ for $x\in\overline{\Omega}$ implies that $w(x, t)>0$ in $Q_T$. Moreover, $c_5(x, t)\equiv 0$ or $c_4(x, t)\displaystyle\int_\Omega c_5(x, y) \mathrm{d}y \leq 1$ on $S_T$, then $w(x, 0)\geq 0$ for $x\in \overline{\Omega}$ implies that $w(x, t)\geq 0$ in $Q_T$.
Lemma 2 Let $\tilde{u}$ and $\bar{u}$ be nonnegative sub-solution and super-solution of $(1.4)$, respectively, with $\tilde{u}(x, 0) \leq \bar{u}(x, 0)$ in $\overline{\Omega}$. Then, $\tilde{u}(x, t)\leq \bar{u}(x, t)$ in $\overline{Q}_T$ if $\tilde{u}>\eta$ or $\bar{u}>\eta$ for some small positive constant $\eta$ holds.
Local in time existence of positive classical solutions of (1.4) can be obtained by using fixed point theorem [13], the representation formula and the contraction mapping principle as in [12], the proof is more or less standard, so is omitted here.
In this section, we shall get some results about the existence and nonexistence of global solution for problem (1.4).
Theorem 1 Assume that $0 < l \leq 1$ and $\displaystyle\int_\Omega g(x, y)\mathrm{d}y < 1$ for all $x\in \partial\Omega$.
(1) If $p+q < m$, then the solution of $(1.4)$ exists globally.
(2) If $p+q=m$, and $a$ is sufficiently small, then the solution of $(1.4)$ exists globally.
(3) If $p+q>m$, and $a$ is sufficiently small, then the solution of $(1.4)$ exists globally provided that $u_0(x)$ satisfies
where $\psi(x), \varepsilon_0$ are defined by (3.1).
Proof Since $\displaystyle\int_\Omega g(x, y)\mathrm{d}y<1$, there exists a sufficiently small constant $\varepsilon_0>0$ such that $\psi(x)$ is the unique solution of the linear elliptic problem
Moreover, $0<\psi(x)<1$ for $x\in\overline{\Omega}$. Denote $ \bar{u}(x, t) = K \psi^{\frac{1}{m}}(x), \ x\in\Omega, t>0$, where $K>1$ is a constant to be determined later. A series of computations yields
(1) If $p+q < m$, we choose that $K$ large enough such that
then we can get
On the other hand, we have for $x\in \partial\Omega, t>0$,
Then that $\bar{u}(x, t)$ is a super-solution of (1.4). Therefore, the solution of (1.4) exists globally.
(2) If $p+q=m$, set $a_0 = \varepsilon_0 |\Omega|^{-1}$, then (3.2)-(3.4) hold provided that $a \leq a_0$ and $K ( >1)$ is large enough. Thus, the solution $u(x, t)$ of (1.4) exists globally.
(3) If $p+q>m$, we can choose $a$ is sufficiently small such that
Then it is easy to verify that (3.2)-(3.4) hold provided $ u_0(x) \leq \big(\frac{\varepsilon_0}{a|\Omega|}\big)^{\frac{1}{p+q-m}} \psi^{\frac{1}{m}}(x)$. Therefore, $\bar{u}(x, t) = K_0 \psi^{\frac{1}{m}}(x)$ is a super-solution of (1.4), thus, the solution $u(x, t)$ of (1.4) exists globally.
Theorem 2 Assume that $l>1$ and $\int_\Omega g(x, y)\mathrm{d}y<1$ for all $x\in \partial\Omega$.
(1) If $p+q=m$, $a$ and $u_0(x)$ are sufficiently small, then the solution of $(1.4)$ exists globally.
(2) If $p+q>m$, and $u_0(x)$ is sufficiently small, then the solution of $(1.4)$ exists globally.
The proof is similar to Theorem 1, here we omit them.
Theorem 3 Assume that $l \geq 1$ and $\displaystyle\int_\Omega g(x, y)\mathrm{d}y <1$ for all $x\in \partial\Omega$. If $p+q>m$, then the solution of $(1.4)$ blows up in finite time provided that $u_0(x)$ is large enough.
Proof To prove the blow-up result, denote by $\varphi(x)$ the first eigenfunction of the problem
with the first eigenvalue $\lambda_1>0$. Let $ K_1 = \min\limits_{x\in \bar{\Omega}} \varphi(x), \ K_2 = \max\limits_{x\in \bar{\Omega}} \varphi(x)$, and $s(t)$ be the solution of the following ODE
where $b>0$ and $\delta, \mu>1$ to be fixed later. Clearly, $s(t) \geq \delta>1$ becomes unbounded in finite time $T(\delta)$. Set $\tilde{u}(x, t) = s^l(t) \varphi^l(x)$, a series of computations yields
Since $p+q>m$, thus, if we choose
Then, from (3.7) and (3.8) we have
On the other hand, if $l=1$, then we have
Second, if $l>1$, by Hölder inequality,
Thus, for $(x, t)\in \partial\Omega \times (0, T(\delta))$ and together with $\int_\Omega g(x, y)\mathrm{d}y<1$, $s(t)>1$ and Jensen's inequality, we obtain
Also for $x\in \overline{\Omega}$, we have
Inequalities (3.9)-(3.11) imply that solution $u(x, t)$ of (1.4) blows up in finite time.
Theorem 4 Assume that $l \geq 1$ and $\int_\Omega g(x, y)\mathrm{d}y \geq 1$ for all $x\in\partial\Omega$. If $p+q>1$, then the solution of $(1.4)$ blows up in finite time provided that $u_0(x)$ is large enough.
Proof Since $p+q>1$, there exists a constant $\mu_0$ such that $ p+q>\mu_0>1$. Let $\tilde{u}(x, t)=s(t)$, which $s(t)$ is a solution of (3.6) with $ b=1, \ \mu=\mu_0, \ \delta = \max\{ 1, \ (a |\Omega| )^{\frac{1}{\mu_0-p-q}} \}$, moreover, $s(t)\geq \delta \geq 1$. For $x\in \Omega, t>0$, computing directly shows
For $x\in\partial\Omega, t>0$, we obtain
Therefore, $\tilde{u}(x, t)$ is a sub-solution of (1.4) provided that $\tilde{u}(x, 0)=s(0)=\delta \leq u_0(x)$, which imply that the solution of (1.4) blows up in finite time.
In this section, we consider the problem (1.4) with $l=1, p+q>m$ and $\displaystyle\int_\Omega g(x, y)\mathrm{d}y<1$ for $x\in\partial\Omega$. By introducing some transformations $v=u^m, t=m \tau$, then (1.4) becomes
where $0 < r=(m-1)/m < 1, p_1=p/m, q_1=q/m$. Under this transformation, assumptions (H1) and (H3)-(H4) become
(G1) $v_0(x)\in C^{2+\alpha}(\Omega)\cap C(\overline{\Omega}), \alpha \in (0, 1), v_0(x)>0$ in $\Omega$, $v_0(x)=\displaystyle\int_\Omega g(x, y) v_0^l(y)\mathrm{d}y$ on $\partial\Omega$.
(G2) $\Delta v_0 + a v_0^{p_1} \int_\Omega v_0^{q_1} \mathrm{d}x>0, x\in\Omega$.
(G3) There exists a constant $\rho \geq \rho_1$ such that
Suppose that the solution $v(x, \tau)$ of (4.1) blows up at finite time $T^*$ and set
then we have the following:
Lemma 3 Suppose that assumptions (H2) and (G1)-(G3) hold, then there exists a positive constant C$_1$ such that
Proof From (4.1), we have (see [14])
Integrating above inequality over $(\tau, T^*)$, then we draw the conclusion.
Lemma 4 Under the conditions of Lemma 3, then there exists a constant $\rho \geq \rho_1$, which is defined in (G3), such that
Proof Let $J(x, \tau) = v_\tau - \rho v^{r+p_1+q_1}$ for $(x, \tau)\in \Omega \times (0, T^*)$, a series of computations yields
By the Hölder inequality, we have
Noting that $ \frac{r+p_1+q_1-1}{r+p_1+2q_1-1} + \frac{q_1}{r+p_1+2q_1-1} =1$, for any $\varepsilon>0$, by virtue of Young's inequality, we obtain
Now, we take
Therefore, it follows from (4.4)-(4.6) that
In addition, for $(x, \tau)\in \partial\Omega\times(0, T^*)$, from (4.1), we have
Therefore,
On the other hand, according to the Hölder inequality, we obtain
Again by the Hölder inequality,
Hence, (4.8) becomes
In addition, (G3) implies that $J(x, 0)\geq 0$. By Lemma 1 we obtain
Integrating (4.3) over $(\tau, T^*)$, we have
Setting $C_2 = \big( \rho (r+p_1+q_1-1)\big)^{-\frac{1}{r+p_1+q_1-1}}$ and combining (4.10) with (4.2), we obtain the following:
Theorem 5 Suppose that conditions (H1), (G1)-(G3) hold and the solution of $(4.1)$ blows up in finite time $T^*$. Then there exist two positive constants $C_1, C_2$ such that
Substituting $r=(m-1)/m, p_1=p/m, q_1=q/m, v=u^m, \tau=m t$ into (4.10), we get the following:
Corollary 1 Suppose that conditions (H1)-(H4) hold and the solution of $(1.4)$ blows up in finite time. Then there exist two positive constants $C_3, C_4$ such that
where $T=T^*/m$ is the blow-up time of $u(x, t)$ and