We will study the parabolic operator

$ \begin{aligned} \ Lu = {a_{ij}}{u_{{x_i}{x_j}}} + {b_i}{u_{{x_i}}} - au - c{u_t} \end{aligned} $

(1.1)

acting on functions in $ D = \Omega \times [0, T]$, where $\ {a_{ij}}(x, t) \in W_\infty ^1(D), {b_i}, a \in {L_\infty }(D)$, $ c = c(x) \in W_\infty ^1(\Omega )$ and $\Omega$ is a connected bounded subset of $n$-dimensional space.

Using a continuous method, Sigillito exlpored the solution for the heat equation, see [1]. Elcart and Sigillito derived an explicit coercivity inequality $ \left\| {\left| u \right|} \right\| \le {\rm const}{\left\| {{L_a}u} \right\|_0}$ and gave a sufficient condition for the existence and uniqueness of solution to the the second order parabolic, see [2].

Recently, in this area the global diffeomorphism theorem was used to prove the existence and uniqueness of solutions of nonlinear differential equation of certain classes. In addition, many authors were extensively investigated this problem, see Mayer [3], Plastock [4], Radulescu and Radulescu [5], Shen Zuhe [6-7], Zampieri [8]. These theorems may be used for solving nonlinear systems of equation.

Motivated by these results, we shall utilize an interesting tool, the attraction basin to give a new set of sufficient condition for the existence and uniqueness of the second order parabolic boundary value problems in this paper, which can be founded in Section 3. Using our approach it is easy to obtain results of Elcart and Sigillito. Moreover, the methods apply not only to this problem but also to other nonlinear diffierential equations.

In this section, we will state some lemmas which are useful to our results. First, we introduce the basin of attraction.

Lemma 2.1(see [8])  Let $G, F$ be Banach spaces, $D$ an open subset of $G$, ${x_0}\in D\ $and $ f:D\subset G\to F$ be a $ {C^1}$ mapping and a local homeomorphism. Then for any $x \in D$, the path-lifting problem

$ \begin{equation} \label{eq:1} \left\{ \begin{aligned} &f({\gamma _x}(t)) = f({x_0}) + e^{ - t}(f(x) - f({x_0})), &t \in R\cdots, \\ &\gamma _x(0) = x, &{\gamma _x}(t) \in D\cdots \end{aligned} \right. \end{equation} $

(2.1)

has a unique continuous solution $ t \to {\gamma _x}(t)$ defined on the maximal open interval $ {I_x} = ({t_{{x^ - }}}, {t_{{x^ + }}}), -\infty \le {t_{{x^ - }}}, {t_{{x^ + }}} \le + \infty .$ Moreover, the set $ \{(x, t) \in D \times R:t \in {I_x}\}$ is open in $ D \times R$ and the mapping is $ (x, t) \to {\gamma _x}(t)$ continuous.

Definition 2.1[8]  In the setting of Lemma 2.1, the basin of attraction of $ {x_0}$ is the set

$ A = \{ X \in D:t_x^ + = + \infty \}. $

Theorem 2.1[9]  With the above setting, $f$ is a global homeomorphism onto $Y$ if and only if $\gamma _x(t)$ is defined on $R$ for all $x\in A$, namely, $\gamma _x(t)$ can also be extended to $-\infty$.

Lemma 2.2(see [8])  Let $X$ be Banach space, $a, b \in R$ and $p:[a, b] \to X$ is a ${C^1}$ mapping on $[a, b]$. Then $\left\| {p(t)} \right\|$ has derivative $\left\| {p(t)} \right\|^\prime$ almost everywhere and $\left\| {p(t)} \right\|^\prime \le \left\| {p(t)} \right\|$ for $a<t<b$.

Second, the following comparison theorem play an important role to prove the sufficient condition for the existence of a unique solution of problem (1.1).

Let $E$ be an open $(t, x)$-set in ${R^2}$ and $g \in C[E, R]$. Consider the scalar differential equation with an initial condition

$ \begin{equation} \label{eq:2} \left\{ \begin{aligned} &\ u' = g(t, u)\cdots, \\ &\ u({t_0}) = {u_0}\cdots. \end{aligned} \right. \end{equation} $

(2.2)

Assume that there exists a sequence ${t_k}$ such that ${t_0} \le {t_k} \to b$ as $k \to \infty $ and ${u^0} = \mathop {\lim }\limits_{k \to \infty } u({t_k})$ exists. If $g(t, u)$ is bounded on the intersection of $E$ and a neighbourhood of $(b, {u^0})$, then $\mathop {\lim }\limits_{t \to b} u(t) = {u^0}$. If in addition, $g(b, {u^0})$ is defined such that $g(t, u)$ is continuous at $(b, {u^0})$, then $u(t)$ is continuously differentiable on $[{t_0}, b]$ and is a solution of (2.2) on $[{t_0}, b]$ (see [10]). In this case the solution $u(t)$ can be extended as a solution to the boundary of $E$.

Theorem 2.2(Comparison theorem in [9])  With the above setting, suppose that $[{t_0}, {t_0} + b)$ is the largest interval in which the maximal solution $r(t)$ of (2.2) exists. Let

$ m \in C[[{t_0}, {t_0} + b), R], (t, m(t)) \in E{\rm{~~for~~}}t \in [{t_0}, {t_0} + b), m({t_0}) \le {u_0} $

and for a fixed Dini derivative

$ Dm(t) \le g(t, m(t)), t \in [{t_0}, {t_0} + b)\backslash T, $

where $T$ denotes an almost countable subset of $t \in [{t_0}, {t_0} + b)\backslash T$, then

$ m(t) \le r(t), t \in [{t_0}, {t_0} + b). $

Consider the boundary value problem

$ \left\{ {_{{u_t} = 0, (x, t) \in \partial \Omega \times (0, T) \cdots, }^{{a_{ij}}{u_{{x_i}{x_j}}} + {b_i}{u_{{x_i}}}- c{u_t}- au = 0 \cdots, }} \right. $

(3.1)

where ${u_t} \in {L_2}([0, T], W_2^2(\Omega ))$. Let ${W_0}(D)$ denote the Hilbert space with the norm

$ {({\left\| u \right\|_{2, 1}})^2} = {\nu ^2}\int {_D{{\left| {{D^2}u} \right|}^2}dxdt} + \int {_D(c{u_t}^2)dt}, $

where $\left| {{D^2}u} \right|^2$ represents the sum of the a squares of all the second derivatives with respect to space variables and $\nu $ is positive constant.

The following assumptions are needed later.

A1  the boundary of $\Omega$ is piecewise smooth with nonnegative mean curvature everywhere.

A2   $f:{W_0}(D) \to {L_2}(D)$ is continuous and a bounded function of $t, {x_1}, \cdots, {x_n}, u$.

Define $S = \sup \left| {{b_i} - {{({a_{ij}})}_{{x_j}}}} \right|$, ${a_1} = {\sup _\Omega }a(x)$ and ${a_0} = {\inf _D}a$, then $S < \sqrt \lambda {v^2}$ and ${a_0} > \sqrt \lambda S - \lambda {v^2}$, where $\lambda = \inf \frac{{\displaystyle\int_\Omega {{{\left| {\Delta u} \right|}^2}dx} }}{{\displaystyle\int_\Omega {{u^2}dx} }} > 0$ is the lowest eigenvalue of $ - \Delta$ in $\Omega$.

Elcart and Sigillito gave the following inequality in [2].

Lemma 3.1  If $u \in {W_0}$, then

$ \;{\left\| u \right\|_{2, 1}} \le C{\left\| {Lu} \right\|_0} \cdots, {\rm{ }} $

(3.2)

where

$ \begin{eqnarray*}{C^2} &=& 336 + 204a_1^2{({a_0} + \lambda {\nu ^2} - S\sqrt \lambda )^{ - 2}} + {C_1}[{n^4}{B^2}{\nu ^{-2}} \\ &&+ {\rm{ }}4{\rm{0}}{\beta ^{\rm{2}}} + 82(2S + 2\mu {\gamma _1} + {\gamma _2})].\end{eqnarray*} $

Denote $Mu = {a_{ij}}{u_{{x_i}{x_j}}} + {b_i}{u_{{x_i}}} - c{u_t}$, then $M$ is the linear operator from ${W_0}(D)$ to ${L_2}(D)$. We may express (3.1) in the form

$ Fu = Mu - au = 0. $

For $u, \phi \in {W_0}(D)$, we have

$ F'(u)(\phi ) = M\phi - {a_u}(x, u(t, x))\phi. $

Define

$ \delta (s) = _{\left\| x \right\| \le s}^{\;\max }\left\| {F'{{(x)}^{ - 1}}} \right\|. $

Theorem 3.1  In the setting of the above, equation (1.1) exists a unique solution if the following conditions hold

(1) $\mathop {\inf }\limits_{\Omega \times R} {a_u}>- \lambda $;

(2) for each, the maximum solution of the initial value problem

$ \begin{aligned} &y'(t) = \eta \delta (y(t)), t \in [0, c)\cdots, \\ &y(0) = 0 \end{aligned} $

(3.3)

is defined on $[0, c)$ and there exists a sequence ${t_n} \to c$ as $n \to \infty $ such that $\mathop {\lim }\limits_{n \to \infty } y({t_n}) = {y^*}$ is finite.

Proof  We have from (2.1) and Lemma 2.2 that

$ \begin{eqnarray*} D\left\| {{\gamma _x}(t)} \right\| &\le& \left\| {{\gamma _x}^\prime (t)} \right\| = \left\| {F'{{({\gamma _x}(t))}^{- 1}}} \right\|{e^{- h}}\left\| {F(x)- F({x_0})} \right\| \\ {\rm{ }} &\le& k{e^{ - h}}\delta (\left\| {{\gamma _x}(t)} \right\|)~~~(k = \left\| {F(x) - F({x_0})} \right\|). \end{eqnarray*} $

By assumption A2, we know the maximum solution $y(t)$ of (3.3) is defined on $[0, c)$ and there exists a sequence ${t_n} \to c$ as $n \to \infty $ such that

$ \mathop {\lim }\limits_{n \to \infty } y({t_n}) = {y^*} $

is finite. It follows that $y(t)$ is continuous on $[0, c)$ and there is a constant $M$ such that

$ \left| {y(t)} \right| \le M, t \in [0, c]. $

By the comparison theorem, we have

$ \left\| {{\gamma _x}(t)} \right\| \le \left| {y(t)} \right| \le M, t \in [0, c]. $

From conditions A1, A2 and condition (1), since $\lambda = \inf \frac{{\displaystyle\int_\Omega {{{\left| {\Delta u} \right|}^2}dx} }}{{\displaystyle\int_\Omega {{u^2}dx} }}>0$ is the lowest eigenvalue of $-\Delta $ in $\Omega$, it follows that for all $u \in {W_0}(D)$, zero is not an eigenvalue of $M\phi -{a_u}(x, u(t, x))\phi$, so for every $u \in {W_0}(D)$, the operator $F'(u) = M -{a_u}I$ is invertible and $F$ is a local homeomorphism from ${W_0}(D)$ onto ${L_2}(D)$, where $I$ denotes the identical operator.

Then in view of Theorem 2.1, we need only show that for all $x \in A$, ${\gamma _x}(t)$ can also be extended to $- \infty $. Namely, we need consider the problem in the opposite direction.

Let $g( - h) = {\gamma _x}(t), t \in (a, 0], h \in [0, -a), a < 0$, for ${t_1}, {t_2} \in (a, 0]$, we have

$ \begin{eqnarray*} \left\| {{\gamma _x}({t_1}) -{\gamma _x}({t_2})} \right\| &=& \left\| {g( -{h_1}) -g( - {h_2})} \right\| = \int_{ - {h_2}}^{ - {h_1}} {\left\| {g'(s)} \right\|} ds \\ &=& \int_{ - {h_2}}^{ - {h_1}} {\left\| {F'{{({\gamma _x}(s))}^{ - 1}}} \right\|} {e^s}\left\| {F(x) - F({x_0})} \right\|ds \\ &\le& \int_{ - {h_2}}^{ - {h_1}} {\left\| {\delta ({\gamma _x}(s))} \right\|} {e^s}\left\| {F(x) - F({x_0})} \right\|ds \\ &\le& k\int_{ - {h_2}}^{ - {h_1}} {\left\| {\delta (y(s))} \right\|} {e^s}ds \\ &\le& k\delta (M){e^{ - a}}\left| {{t_1} - {t_2}} \right|. \\ \end{eqnarray*} $

So ${\gamma _x}(t)$ is Lipschitz continuous on $\left( { - \mathit{a}{\rm{,}}} \right.\left. 0 \right],{\mathit{\gamma }_\mathit{x}}\left( \mathit{t} \right)$ can also be extended to $ - \infty $, the theorem is proved.

Elcart and Sigillito [2] have studied the following initial-boundary value problem

$ \begin{aligned} &a_{ij}{u_{{x_i}{x_j}}} + {b_i}{u_{{x_i}}} - c{u_t} = f(x, t, u)\cdots, \\ &u = 0{\rm{~~ for ~~ t}} = {\rm{0~~ and ~~(x, t)}} \in \partial \Omega \times {\rm{(0, T)}}\cdots, \end{aligned} $

(3.4)

where $\partial \Omega \in {C^2}$ and $\mathit{f}$ is continuous and has three derivatives with respect to $\mathit{u}$. The problem (3.4) may be formulated as an operator equation $Pu = 0$, where $Pu = Mu - f(x, u)$is a mapping of ${W_0}(D)$ onto ${L_2}(D)$.

Corollary 3.1  Assume that $\mathit{f}$ satisfies

(ⅰ) $\mathop{\inf }\limits_{\Omega \times R} {f'_u}>- \lambda $;

(ⅱ) Uniformly in x, $\left\| {{{f'}_u}} \right\| = \omega (\left\| u \right\|)$, where $\omega $ is continuous map satisfying $\displaystyle\int_a^\infty {\frac{{dt}}{{\omega (t)}}} = \infty$, Then there is a unique solution of the equation $Pu = 0$ in ${W_0}(D)$.

Proof  Compare with equations (3.4) and (3.1), we have $f(x, u) = au$, so condition (1) of Theorem 3.1 is satisfied. Denote ${\omega _1}(t) =\alpha \omega (t) + \beta, $ then $\displaystyle\int_a^\infty {\frac{{dt}}{{{\omega _1}(t)}}} = \infty$ and $\delta (s) = _{\left\| x \right\| \le s}^{\;\max }\left\| {P'{{(x)}^{ - 1}}} \right\|.$ We have from Lemma 3.1

$ \begin{aligned} \left\| {{{[P'(u)]}^{ - 1}}} \right\| \le \alpha \mathop {\sup }\limits_\Omega \left| {{{f'}_u}(x, u(x))} \right| + \beta \cdots \end{aligned} $

(3.5)

for positive constant $\alpha, \beta $, then $\delta (t) \le {\omega _1}(t), $ and thus

$ \begin{aligned} \int_a^{ + \infty } {\frac{{dt}}{{\delta (t)}}} \ge \int_a^{ + \infty } {\frac{{dt}}{{{\omega _1}(t)}}} = + \infty \cdots. \end{aligned} $

(3.6)

From problem (3.3), $\forall t > 0$, $\left| {\int_0^\mathit{t} {\frac{{\mathit{y'}(\mathit{r})}}{{\mathit{\delta }(\mathit{y}(\mathit{r}))}}\mathit{dr}} } \right| = \mathit{\eta t}$. Let $y(r) = s$, we have $\displaystyle\int_{{\rm{y(0)}}}^{y({\rm{t)}}} {\frac{1}{{\delta (s)}}} ds = \eta t$.

For equation (3.6), we have that $y(t)$ is bounded. Consequently, there exist is a real sequence $\{t_n\}$: ${t_n} \to c$ as $n \to \infty $ such that $\mathop {\lim }\limits_{n \to \infty } y({t_n}) = {y^*}$ exists. The corollary is proved.

Remark  Condition (ⅱ) in Corollary 3.1 can be replaced with ${f'_u} = {\rm O}(u)$, because $\displaystyle\int_a^\infty {\frac{{dt}}{{\omega (t)}}} = \infty $ holds.The result of Elcart and Sigillito in [2] becomes a special case of the theorem 3.1.