MATHEMATICS

Yu. A. DUBINSKII

ON NONLINEAR PARABOLIC EQUATIONS HAVING NONDIVERGENT FORM

(Presented by Academician I. G. Petrovskii on January 4, 1965)

Let \(Q=\{a\le x\le b,\ 0\le t\le T\}\) be a rectangle with lateral boundary \(\Gamma\); \(u(x,t)\) a function defined in \(Q\); \(u^{(s)} \equiv d^s u/dt^s\), \(D^l u \equiv \partial^l u/\partial x^l\). We shall consider known the following spaces:
\[ W_p^{(l)}(a,b)\equiv W_p^{(l)},\quad \dot W_p^{(l)}(a,b)\equiv \dot W_p^{(l)},\quad \dot W_{p'}^{(-l)}(a,b)\equiv \dot W_{p'}^{(-l)} \quad (p>1,\ p'=p/(p-1)). \]
Further, \(W_{p,q}^{(l,1)}\) denotes the subspace of \(W_{p,q}^{(l,1)}\) such that, for
\(u(x,t)\in W_{p,q}^{(l,1)}\),
\[ u(x,0)=0,\quad u|_{\Gamma}=0,\ldots,\ D^{l-1}u|_{\Gamma}=0; \]
\(\dot W_{p,q}^{(l,1)}\) is the same, except that \(u(x,T)=0\) \((q\ge1)\). The norms of the spaces
\(\mathcal L_p(0,T;\dot W_p^{(l)})\) and
\(\mathcal L_p(0,T;\dot W_{p'}^{(-l)})\) will be denoted by
\(\|u\|_{l,p}\) and \(\|u\|_{-l,p'}\).

Let \(k\ge0,\ l\ge0,\ m\ge1\) be integers, with \(k+l=2m\). Then:

a) \(H=H(l,k,r,p,p')\) is the closure of smooth functions \(u(x,t)\),
\[ u(x,0)=0,\quad u|_{\Gamma}=0,\ldots,\ D^{r-1}u|_{\Gamma}=0,\quad D^{2r}u|_{\Gamma}=0,\ldots,\ D^{l-1}u|_{\Gamma}=0 \]
in the norm
\[ \|u\|=\|u\|_{l,p}+\|u'\|_{-k,p'}, \]
where \(k\le l,\ 2r=l-k\);

b) \(H_1=H_1(l,k,r,p,p')\) is the space of functions \(u(x,t)\) of the form
\[ u(x,t)=D^{2r}v(x,t),\quad v(x,t)\in\mathcal L_p(0,T;\dot W_p^{(k)}),\quad u'(x,t)\in\mathcal L_{p'}(0,T;\dot W_{p'}^{(-k)}), \]
where \(k\ge l,\ 2r=k-l\).

Finally, we note that constants will be denoted by the letter \(K\).

§ 1. Nonlinear strongly parabolic equations.

The case \(k\le l\). Put \(l-k=2r\), and in the rectangle \(Q\) consider the problem
\[ \mathcal L(u)\equiv (-1)^r u' + D^k A(x,t,D^l u)=h(x,t); \tag{1} \]
\[ u(x,0)=0,\quad u|_{\Gamma}=0,\ldots,\ D^{r-1}u|_{\Gamma}=0,\quad D^{2r}u|_{\Gamma}=0,\ldots,\ D^{l-1}u|_{\Gamma}=0 * . \tag{2} \]

Assumptions. I. The function \(A(x,t,\xi)\) is continuous in all arguments for \((x,t)\in Q\) and arbitrary \(\xi\), and
\[ |A(x,t,\xi)|\le K\bigl(|\xi|^{p-1}+1\bigr), \]
where \(p>1\) is a fixed number;
\[ h(x,t)\in\mathcal L_{p'}(0,T;\dot W_{p'}^{(-k)}). \]

II. Parabolicity condition. For every function
\[ u(x,t)\in\mathcal L_p(0,T;W_p^{(l)}) \]
and every \(t\in[0,T]\) the inequality
\[ (-1)^k\bigl(A(x,t,D^l u),D^l u\bigr)\ge a_0\bigl(\|D^l u\|^{p-1},\ \|D^l u\|\bigr)-K(t) \]
holds, where \(a_0>0\), \(K(t)\) is a continuous function on \([0,T]\), and
\[ (u,v)=\int_a^b uv\,dx . \]

III. Definiteness condition (strong parabolicity). For any \(u(x,t)\) and \(v(x,t)\) from
\(\mathcal L_p(0,T;W_p^{(l)})\) the inequality
\[ (-1)^k\bigl[A(x,t,D^l u)-A(x,t,D^l v),\ D^l(u-v)\bigr]\ge \]
\[ \ge a_1\bigl[\|D^l(u-v)\|^{p-1},\ \|D^l(u-v)\|\bigr]; \]
\[ a_1>0,\quad [u,v]=\int_Q uv\,dx\,dt . \]

* The case of nonzero conditions that admit extension into \(Q\) from the space \(H(H_1)\) is treated analogously.

Theorem 1. If conditions I–III are satisfied, then the mapping
\(\mathcal L: H \to \mathcal L_{p'}(0,T;\dot W_{p'}^{(-k)})\) is a homeomorphism.

Proof. The most essential point in the proof of this theorem is the solvability of problem (1), (2). To prove this fact, choose a system of smooth functions \(v_1(x), v_2(x),\ldots,\) complete in \(\dot W_p^{(k)}\). We shall seek an approximate solution \(u_n(x,t)\) from the relation \(z_n(x,t)=D^{2r}u_n(x,t)\) under the condition
\[ u_n|_\Gamma=0,\ldots,\quad D^{r-1}|_\Gamma=0, \]
where
\[ z_n=\sum_{\nu=1}^n c_{\nu n}(t)v_\nu(x). \]
The unknown functions \(c_{\nu n}(t)\) are determined from the system of ordinary differential equations
\[ (-1)^r(u_n',v_\nu)+(-1)^k(A(x,t,D^l u_n),D^k v_\nu),\quad c_{\nu n}(0)=0 \]
\[ (\nu=1,\ldots,n). \tag{3} \]

Lemma. Let \(c=(c_1(t),\ldots,c_n(t))\), and let the vector function \(\mathbf f(t,c)\) be continuous in \(t\) and \(c\). Suppose, further, that for any \(c\) and any \(t\in[0,T]\) the inequality
\[ \int_0^t \mathbf f(\tau,c)c\,d\tau \ge -\left(a_2\int_0^t |c|^2\,d\tau+K(t)\right), \]
holds, where \(a_2\ge0\), \(K(t)\ge0\) is a continuous function on \([0,T]\). Then the system
\[ c'+\mathbf f(t,c)=0,\qquad c(0)=0 \]
has at least one solution defined on the whole interval \([0,T]\).

From this lemma (with a suitable choice of \(v_1(x),v_2(x),\ldots\)) there follows the solvability of system (3). For the approximate solutions \(u_n(x,t)\) the a priori estimate
\[ [|D^l u_n|^{p-1},\,|D^l u_n|]\le K \]
is valid. From this estimate it follows that there exists a function \(u(x,t)\in\mathcal L(0,T;W_p^{(l)})\) and a subsequence (which we again denote by \(u_n(x,t)\)) such that
\[ u_n\to u,\ldots,\quad D^l u_n\to D^l u \]
weakly in \(\mathcal L_p(Q)\). Further, by virtue of I we may assume that
\[ A(x,t,D^l u_n)\to a(x,t) \]
weakly in \(\mathcal L_{p'}(Q)\). It is easy to see that the function \(u(x,t)\) satisfies the integral identity
\[ (-1)^{r+1}[u,v']+(-1)^k[a(x,t),D^k v]=[h,v], \quad \forall v\in\dot W_{p,2}^{(k,1)}. \tag{4} \]

Putting \(v=\psi(x)\varphi(t)\), where \(\psi\in\dot W_p^{(k)}\), \(\varphi\in \dot C^\infty(0,T)\), and taking into account that \(a(x,t)\in\mathcal L_{p'}(Q)\), we derive from (4) that
\[ u'\in \mathcal L_{p'}(0,T;\dot W_{p'}^{(-k)}), \]
i.e. \(u\in H\). Hence it follows that \(u(x,t)\) satisfies the relation
\[ (-1)^r[u',v]+(-1)^k[a(x,t),D^k v]=[h,u], \quad \forall v\in\mathcal L_p(0,T;\dot W_p^{(k)}). \]
Comparing these relations, we see that \(u(x,0)=0\).

It remains to prove that
\[ D^k a(x,t)=D^k A(x,t,D^l u). \]
For this purpose set
\[ W_N=\{w(x,t),\ |w|_\Gamma=0,\ldots,\ D^{r-1}w|_\Gamma=0,\ D^{2r}w=\sum_{\nu=1}^N \alpha_\nu(t)v_\nu(x)\}, \]
where \(\alpha_\nu(t)\in C^\infty(0,T)\), \(N\ge1\) is an integer, and use condition III, according to which for any function \(w(x,t)\in W_N\) we have
\[ (-1)^r[u_n'-w',D^{2r}(u_n-w)]+ \]
\[ +(-1)^k(A[x,t,D^l u_n)-A(x,t,D^l w),D^l(u_n-w)]\ge0. \tag{5} \]
Now note that from (3) it obviously follows that \(u_n(x,t)\) satisfies the relation
\[ (-1)^r[u_n',v]+(-1)^k[A(x,t,D^l u_n),D^k v]=[h,v], \quad \forall v=\sum_{\nu=1}^n \alpha_\nu(t)v_\nu(x). \]
In particular, for \(w\in W_N\) with \(N\le n\) we obtain
\[ (-1)^r[u_n',D^{2r}(u_n-w)] +(-1)^k[A(x,t,D^l u_n),D^l(u_n-w)] \]
\[ =[h,D^{2r}(u_n-w)], \]
which together with (5) gives the inequality
\[ [h,D^{2r}(u_n-w)] +(-1)^{r+1}[w',D^{2r}(u_n-w)] + \]
\[ +(-1)^{k+1}[A(x,t,D^l w),D^l(u_n-w)]\ge0. \tag{6} \]
Since (6) contains no nonlinearities in \(u_n\), as \(n\to\infty\) we obtain
\[ [h,D^{2r}(u-w)] +(-1)^{r+1}(w',D^{2r}(u-w)] + \]
\[ +(-1)^{k+1}[A(x,t,D^l w),D^l(u-w)]\ge0. \]

Further, since \(D^{2r}(u-w)\in \mathscr L_p(0,T;\dot W_p^{(k)})\), then, taking (5) into account, we find

\[ (-1)^r[u'-w',D^{2r}(u-w)]+ +(-1)^k[a(x,t)-A(x,t,D^l w),D^l(u-w)]\ge 0. \]

The last inequality has been proved for \(w\in W_N\); but since the linear combinations \(\sum_{\nu=1}^{N}a_\nu(t)v_\nu(x)\) are strongly dense in \(\mathscr L_p(0,T;\dot W_p^{(k)})\), it is valid for any function \(w(x,t)\) \((w|_\Gamma=0,\ldots,D^{r-1}w|_\Gamma=0)\), for which \(D^{2r}w\in \mathscr L_p(0,T;\dot W_p^{(k)})\). In particular, putting \(w=u-\xi\zeta\), \(\zeta\in H\), \(\xi\to +0\), we obtain that

\[ (-1)^k[a(x,t)-A(x,t,D^l u),D^l\zeta]\ge 0. \tag{7} \]

The latter is possible only in the case when \(D^k a(x,t)=D^k A(x,t,D^l u)\).* The uniqueness of the solution follows from condition III, and from this also follows the continuity of the inverse operator \(\mathscr L^{-1}\). The theorem is proved.

Example. \(\mathscr L(u)\equiv (-1)^r u'+(-1)^kD^k(|D^l u|^{p-1}\operatorname{sign}D^l u)\); all the conditions of Theorem 1 for the operator \(\mathscr L(u)\) are fulfilled. Let us note the special case \(k=0,\ l=2m\). In this case
\[ \mathscr L(u)\equiv (-1)^m u'+|D^{2m}u|^{p-1}\operatorname{sign}D^{2m}u \]
under the conditions of the first boundary-value problem maps \(\dot W_{p,p}^{(2m,1)}\) homeomorphically onto \(\mathscr L_p\).

Theorem 2. If \(k\ge l\) and conditions I–III are fulfilled, then the mapping
\[ \mathscr L:H_1\to \mathscr L_p'(0,T;\dot W_p^{(-k)}) \]
is a homeomorphism.

§ 2. General nonlinear parabolic equations.

The case \(k_0\le l_0\). Let \((k,l,s,n)\) be some collection of sets of nonnegative integers, where \(k+l=2m\), \(s+n\le 2m-1\), \(n\le l_0-1\), \(l_0=\max l\); \(k_0=2m-l_0\), \(l_0-k_0=2r_0\). For a given collection of sets we study the problem

\[ (-1)^{r_0}u'+\mathscr L(u)+T_1(u)+T_2(u)\equiv (-1)^{r_0}u'+\sum_{k+l=2m}D^k A_k(x,t,u,\ldots,D^l u)+ \]
\[ +\sum_{\substack{k+l\le 2m-1,\ s\ge k_0}}D^s B_s(x,t,u,\ldots,D^n u)+ \]
\[ +\sum_{\substack{k+l\le 2m-1,\ s<k_0}}D^s B_s(x,t,u,\ldots,D^n u)=0, \tag{8} \]

\[ u(x,0)=0,\quad u|_\Gamma=0,\ldots,D^{r_0-1}u|_\Gamma=0, \qquad D^{2r_0}u|_\Gamma=0,\ldots,D^{l_0-1}u|_\Gamma=0. \tag{9} \]

Assumptions. \(I'\). The functions \(A_k(x,t,\xi^0,\ldots,\xi^l)\), \(B_s(\ldots)\) are continuous in all arguments, and

\[ |A_k(x,t,\xi^0,\ldots,\xi^l)|\le K\prod_{i=1}^{l}\bigl(|\xi^i|^{p_i}+1\bigr),\qquad |B_s(x,t,\xi^0,\ldots,\xi^n)|\le \]
\[ \le K\prod_{i=1}^{n}\bigl(|\xi^i|^{q_i}+1\bigr), \]

where \(p_i\ge 0,\ q_i\ge 0,\ \sum p_i\le p-1,\ \sum q_i\le p-1\). In addition, suppose that for \(k>k_0\) \((s>k_0)\) the functions \(A_k(x,t,\xi^0,\ldots,\xi^l)\) \((B_s(\ldots))\) are differentiable with respect to \(x,t\) and \(\vec \xi\) \(k-k_0\) \((s-k_0)\) times, and each differentiation with respect to \(x\) and \(t\) does not increase the growth in \(\xi\), while each differentiation with respect to \(\xi^i\) \((i=1,\ldots,n;\,l)\) decreases the growth in \(\xi^i\) by one. We introduce the notation:

\[ \mathscr L(u,v)\equiv (-1)^{k_0}\sum_{k+l=2m}[D^{k-k_0}A(x,t,u,\ldots,D^l u),D^{k_0}v], \]
and \(T_1(u,v)\) and \(T_2(u,v)\) are defined analogously.

\(II'\). Parabolicity condition. For any function
\[ u(x,t)\in \mathscr L_p(0,T;W_p^{(l_0)}) \]
\[ \mathscr L_0(u,D^{2r_0}u)\equiv \mathscr L(u,D^{2r_0}u)+ \]
\[ +T_1(u,D^{2r_0}u)+T_2(u,D^{2r_0}u)\ge a_0[|D^{l_0}u|^{p-1},|D^{l_0}u|]-K;\quad a_0>0. \]

* Let us note that the idea of monotonicity was used earlier by a number of authors: M. M. Vainberg and R. I. Kachurovskii, J. Minty, F. Browder, and others.

III′. The condition of semiboundedness of the “variation.” For any \(u \in \mathscr L_p(0,T; W_p^{(l_0)})\) and \(\vartheta \in \mathscr L_p(0,T; W_p^{(l_0)})\) from the ball \(\|u\|_{l_0,p}\le R\), the inequality
\[ \mathscr L_0(u,D^{2r_0}(u-v))-\mathscr L_0(v,D^{2r_0}(u-v)) \ge -c(R,\|u-v\|_{l_0-1,p}) \]
holds, where \(c(R,\rho)\ge 0\) is a continuous function such that, for any \(R\) and \(\rho\),
\(c(R,\xi\rho)/\xi \to 0\) as \(\xi\to +0\).

Theorem 3. If conditions I′—III′ are satisfied, then problem (8), (9) has at least one solution in the space
\[ H=H(l_0,k_0,r_0,p,p'). \]

Proof. First we shall have to solve an elliptic equation containing a small parameter \(\varepsilon>0\), namely the equation
\[ (-1)^{r_0+1}\varepsilon u''+(-1)^r u' + \mathscr L(u)+T_1(u)+T_2(u)=0 \tag{10} \]
with the additional condition \(u'(x,T)=0\). We seek an approximate solution of this problem from the relation \(D^{2r_0}u_{n\varepsilon}=z_{n\varepsilon}\), under the conditions \(u_{n\varepsilon}|_\Gamma=0,\ldots,\)
\[ \ldots,\quad D^{r_0-1}u_{n\varepsilon}|_\Gamma=0, \]
where
\[ z_{n\varepsilon}(x,t)=\sum_{\nu=1}^n c_{\nu n}v_\nu(x,t); \]
\(v_1(x,t), v_2(x,t),\ldots\) is a system of smooth functions, complete in \(\mathring W_{p,2}^{(l_0,1)}\). The unknown constants \(c_{\nu n}\) are determined from the system of equations
\[ (-1)^{r_0}\varepsilon [u_{n\varepsilon}',v_\nu'] +(-1)^{r_0}[u_{n\varepsilon}',v_\nu] +\mathscr L(u_{n\varepsilon},v_\nu) +T_1(u_{n\varepsilon},v_\nu)+T_2(u_{n\varepsilon},v_\nu)=0, \]
\[ \nu=1,\ldots,n. \]

The solvability of this system follows from the lemma of M. I. Vishik (1); moreover, the following a priori estimate is valid:
\[ \varepsilon [D^{r_0}u_{n\varepsilon}',D^{r_0}u_{n\varepsilon}'] +\bigl[|D^{l_0}u_{n\varepsilon}|^{p-1},|D^{l_0}u_{n\varepsilon}|\bigr]\le K. \]
From this estimate and J. Aubin’s theorem (2) it follows that, for fixed \(\varepsilon>0\), there exists a function \(u_\varepsilon(x,t)\in \mathscr L_p(0,T;W_p^{(l_0)})\) and a subsequence (denote it again by \(u_{n\varepsilon}(x,t)\)) such that \(u_{n\varepsilon}\to u_\varepsilon\) \((n\to\infty)\) strongly in \(\mathscr L_p(0,T;W_p^{(l_0-1)})\), and, moreover, \(D^{l_0}u_{n\varepsilon}\to D^{l_0}u_\varepsilon\) weakly in \(\mathscr L_p(Q)\), while \(D^{r_0}u_{n\varepsilon}'\to D^{r_0}u_\varepsilon'\) weakly in \(\mathscr L_2(Q)\). As in § 1, we conclude that
\[ (-1)^{r_0}\varepsilon [u_\varepsilon',v'] +(-1)^{r_0}[u_\varepsilon',v] +\mathscr L(u_\varepsilon,v)+T_1(u_\varepsilon,v)+T_2(u_\varepsilon,v)=0, \]
\[ \forall v\in \mathring W_{p,2}^{(k_0,1)}. \]

Hence, putting \(v=\psi(x)\varphi(t)\), \(\psi\in \mathring W_p^{(k_0)}\), \(\varphi\in \mathring C^\infty(0,T)\), we find that
\[ -\varepsilon u_\varepsilon''+u_\varepsilon'=g_\varepsilon(t), \]
where
\[ g_\varepsilon(t)\in \mathscr L_{p'}(0,T;\mathring W_{p'}^{(-k_0)}) \]
and ranges there over a bounded set. Solving this equation under the condition \(u_\varepsilon'(T)=0\), we obtain that
\[ u_\varepsilon'\in \mathscr L_{p'}(0,T;\mathring W_{p'}^{(-k_0)}) \quad\text{and}\quad \|u_\varepsilon'\|_{-k_0,p'}\le K \]
as \(\varepsilon\to 0\). Therefore, again by J. Aubin’s theorem, one may assume that
\[ u_\varepsilon\to u\in H \]
strongly in
\[ \mathscr L_p(0,T;W_p^{(l-1)}), \]
and
\[ D^{l_0}u_\varepsilon\to D^{l_0}u \]
weakly in \(\mathscr L_p(Q)\). We then conclude (again repeating the scheme of § 1) that \(u(x,t)\) is the desired solution of problem (8), (9). The theorem is proved.

Theorem 4. If \(k_0\ge l_0\) and conditions I′—III′ are satisfied, then equation (8) has at least one solution in the space
\[ H_1=H_1(l_0,k_0,r_0,p,p'). \]

§ 3. It is obvious that what was said in §§ 1 and 2 generalizes to the cylinder
\[ Q=G\times[0,T],\quad G\subset R^n, \]
if, formally, in equations (1), (8) one replaces \(D\) by a suitable elliptic operator \(A\). For example, the results of Theorem 1 are valid for the problem
\[ (-1)^m u'+|\Delta^m u|^{p-1}\operatorname{sign}\Delta^m u=h(x,t), \]
\[ u(x,0)=0,\qquad D^\omega u|_\Gamma=0,\qquad \omega=(\omega_1,\ldots,\omega_n), \]
\[ |\omega|=\omega_1+\cdots+\omega_n\le m-1. \]

In conclusion, I take this opportunity to express my gratitude to Prof. M. I. Vishik for his attention to my work.

Moscow Power Engineering Institute

Received
4 I 1965

References

1. M. I. Vishik, Tr. Mosk. matem. obshch., 12 (1963).
2. J. P. Aubin, C. R., 256, 5042 (1963).