K. V. VALIKOV

ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A NONLINEAR PARABOLIC EQUATION

(Presented by Academician I. G. Petrovskii on 11 IX 1964)

In the present note the results of the paper \((^5)\) are applied to the study of the asymptotic behavior of solutions of a parabolic equation. Consider, in a bounded domain \(\Omega\) with \(2m\)-times smooth boundary \(S\) of real \(n\)-dimensional space \(R_n\) \((n \ge 2)\), the elliptic operator

\[ \mathcal L(x,D)=\sum_{|\beta|\le 2m} a_\beta(x)D^\beta,\qquad \beta=(\beta_1,\ldots,\beta_n), \]

\[ |\beta|=\beta_1+\cdots+\beta_n,\quad D^\beta=D_1^{\beta_1}\cdots D_n^{\beta_n},\quad D_i=\partial/\partial x_i. \]

We assume the leading coefficients to be continuous in \(\overline{\Omega}\), the remaining ones measurable and bounded. Let differential operators \(B_j(x,D)\), \(j=1,\ldots,m\), of orders \(m_j<2m\), with coefficients from \(C_{2m-m_j}(S)\), be given on \(S\). We shall assume \(B_j\) to be subject to the complementing condition \((^2)\), and \(\mathcal L\), for \(n=2\), to satisfy the root condition \((^2)\). In addition, the system \(B_j\) will be regarded as normal, i.e. the \(m_j\) are distinct and \(S\) is not characteristic for \(B_j\) at any point. Consider the following problem:

\[ \frac{\partial u(t,x)}{\partial t} = \mathcal L(x,D)u+F(t,x,u,Du,\ldots,D^k u), \qquad t\ge t_0, \tag{1} \]

\[ B_j u=0,\quad j=1,\ldots,m;\qquad u(t_0,x)=u_0(x),\quad 0\le k\le 2m-1. \]

Here \(F(t,x,u,\ldots,D^k u)\) is defined and continuous in the totality of variables for \(t\ge t_0\), \(x\in\overline{\Omega}\), \(-\infty<u,\ldots,D^k u<\infty\), and satisfies, with respect to the variables \(u,\ldots,D^k u\), a Lipschitz condition with constant \(\gamma(t)\); \(\gamma(t)\) is continuous and nonnegative for \(t\ge 0\). Recently obtained results \((^1)\) make it possible to apply to (1) the methods of semigroup theory. The operator \(\mathcal L\) and the boundary operators \(B_j\) generate in \(L_p(\Omega)\) \((p>1)\) a closed linear operator \(A_p\) with domain of definition
\[ D(A_p)=W_p^{2m}(\Omega;B_j), \]
the set of functions from \(W_p^{2m}(\Omega)\) satisfying the boundary conditions in the sense of S. L. Sobolev.

Let the following conditions be fulfilled:

1. \[ (-1)^m\frac{\mathcal L'(x,\xi)}{|\mathcal L'(x,\xi)|}\ne e^{i\theta} \quad\text{for }x\in\overline{\Omega},\ \theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right] \]
   and for any real
   \[ \xi=(\xi_1,\ldots,\xi_n)\ne 0. \]

2. If \(x\in S\), \(\nu\) is the normal to \(S\) at the point \(x\), \(\xi\ne 0\) and is parallel to the tangent at the point \(x\) to \(S\), and \(t_k^+(\xi,\lambda)\) are the \(m\) roots of the polynomial
   \[ (-1)^m\mathcal L'(x,\xi+t\nu)-\lambda \]
   with positive imaginary part, where \(\lambda\) is any number on the ray \(\arg \lambda=\theta\), then the polynomials
   \[ B_j'(x,\xi+t\nu),\qquad j=1,\ldots,m, \]
   are linearly independent modulo the polynomial
   \[ \prod_{k=1}^{m}\bigl(t-t_k^+(\xi,\lambda)\bigr). \]
   Here
   \[ \mathcal L'(x,D),\ B_j'(x,D) \]
   are the principal parts of \(\mathcal L(x,D)\), \(B_j(x,D)\).

From the results of \((^1)\) it follows that, when conditions 1, 2 are satisfied, \(A_p\) generates in \(\mathscr L_p(\Omega)\) an analytic semigroup \(e^{A_p t}\), \(t>0\), strongly continuous at \(t=0\). Moreover \(R(\lambda; A_p)\) is completely continuous, and the spectrum of \(A_p\) is discrete and does not depend on \(p\). Let \(\lambda_0\) be a real number, \(\lambda_0>\sup \operatorname{Re}\sigma(A_p)\), where \(\sigma(A_p)\) is the spectrum of \(A_p\). For the operator \(\lambda_0 I-A_p\), fractional powers \((\lambda_0 I-A_p)^\alpha\), \(\alpha\geqslant 0\), are defined.

Theorem 1. Let \(p>1\), \(k\) be an integer, \(0\leqslant k\leqslant 2m-1\), \((2m-k)p>n\), \(\alpha=\dfrac{k+n/p}{2m}\). In that case, for any \(\varepsilon>0\) the operator
\[ (\lambda_0 I-A_p)^{-\alpha-\varepsilon} \]
acts completely continuously from \(\mathscr L_p(\Omega)\) into \(C_k(\overline\Omega)\).

The method of proof of the theorem is analogous to the method of \((^3)\) and uses one embedding theorem of V. P. Il’in \((^4)\) and an a priori estimate in \(\mathscr L_p(\Omega)\) for solutions of the elliptic equation \((^2)\). The function \(F(t,x,u,\ldots,D^k u)\) generates in \(\mathscr L_p(\Omega)\) a nonlinear operator \(F(t,u)=F(t,x,u,\ldots,D^k u)\); moreover, under the conditions of Theorem 1, \(F(t,u)\) is defined for \(t\geqslant 0\), \(u\in D(A_p^{\alpha+\varepsilon})\), \(\varepsilon>0\), and \(\Phi(t,u)=F(t,A_p^{-\alpha-\varepsilon})\) is defined for all \(t\geqslant 0\), \(u\in \mathscr L_p(\Omega)\), is continuous (in the metric of \(\mathscr L_p(\Omega)\)) with respect to \((t,u)\), and satisfies the condition
\[ \|\Phi(t,u_1)-\Phi(t,u_2)\|_{\mathscr L_p(\Omega)} \leqslant k(\varepsilon)\gamma(t)\|u_1-u_2\|_{\mathscr L_p(\Omega)}. \]

We shall call a solution of the integral equation in the Banach space \(\mathscr L_p(\Omega)\)
\[ u(t)=e^{A_p(t-t_0)}u_0+\int_{t_0}^{t} e^{A_p(t-\tau)}F(\tau,u(\tau))\,d\tau . \tag{2} \]
a generalized solution of problem (1).

Here \(u_0\in \mathscr L_p(\Omega)\), and \(u(t)\) are functions for \(t\geqslant t_0\) with values in \(\mathscr L_p(\Omega)\), i.e. \(u(t)=u(t,x)\), where \(u(t,x)\), for each fixed \(t\), is an element of \(\mathscr L_p(\Omega)\). The existence and uniqueness of the generalized solution for any \(u_0\in \mathscr L_p(\Omega)\) are proved in \((^5)\). The generalized solution is \(k\) times continuously differentiable with respect to \(x\) for \(t>t_0\), \(x\in \Omega\).

Let \(\lambda_s\), \(s=1,2,\ldots\), be the eigenvalues of the operator \(A_p\); \(\mu_s=\operatorname{Re}\lambda_s\); let \(d_s\) be all the distinct values among the \(\mu_s\), numbered so that \(d_s>d_{s+1}\). The only possible accumulation point of the \(d_s\) is \(-\infty\). Denote by \(\mathscr L(\{d_j\}_{j=1}^r)\) the subspace of \(\mathscr L_p(\Omega)\) consisting of the eigenvectors and associated vectors of the operator \(A_p\) corresponding to eigenvalues \(\lambda\) with \(\operatorname{Re}\lambda=d_j\) for some \(j=1,\ldots,r\). Then \(\mathscr L_p(\Omega)\) can be represented as the direct sum of two subspaces reducing \(A_p\):
\[ \mathscr L_p(\Omega)=M(\{d_j\}_{j=1}^r)+\mathscr L(\{d_j\}_{j=1}^r). \]

Definition 1. A transformation \(f\) acting from \(M(\{d_j\}_{j=1}^r)\) into \(\mathscr L_p(\Omega)\) will be called regular if it is continuous and, for any \(u\in M(\{d_j\}_{j=1}^r)\), the projection \(fu\) onto \(M(\{d_j\}_{j=1}^r)\) coincides with \(u\). The image \(M(\{d_j\}_{j=1}^r)\) will be called a regular image of \(M(\{d_j\}_{j=1}^r)\).

Definition 2. Let \(u(t,x)\) be defined for sufficiently large \(t\) and \(x\in\Omega\). We shall call the exponent of \(u\) in \(\mathscr L_p(\Omega)\)
\[ \omega_p(u)=\lim_{t\to\infty}\frac{\ln\|u(t,x)\|_{\mathscr L_p(\Omega)}}{t}. \]
Analogously, the exponent of \(u\) in \(C_k(\overline\Omega)\) is
\[ \omega_{C_k}(u)=\lim_{t\to\infty}\frac{\ln\|u(t,x)\|_{C_k(\overline\Omega)}}{t}. \]
We assume that the right-hand sides of the equalities make sense.

Theorem 2. Let an integer \(r\geqslant 1\), \(\varepsilon>0\),
\[ \varepsilon<\min_{1\leqslant i\leqslant r}\frac{d_i-d_{i+1}}{2}, \]
and numbers \(p,k,m,n\) be given satisfying the conditions of Theorem 1,
\[ \alpha\in\left(\frac{k+n/p}{2m},\,1\right). \]

There exists a \(\delta=\delta(\varepsilon,r,\alpha)>0\) such that, if \(\gamma(t)\) satisfies the condition

\[ \int_{t_0}^{t}\frac{e^{(\varepsilon/2)(\tau-t)}\gamma(\tau)}{(t-\tau)^\alpha}\,d\tau,\quad \int_t^\infty e^{(\varepsilon/2)(t-\tau)}\gamma(\tau)\,d\tau<\delta,\quad t>t_0, \]

then for any generalized solution of problem (1) either \(\omega_p(u)<d_{r+1}+\varepsilon\), or \(\omega_p(u)\) lies in one of the \(\varepsilon\)-neighborhoods of the points \(d_i\), \(i=1,\ldots,r\). Moreover, \(\omega_p(u)=\omega_{C_k}(u)\), and the set of initial functions \(u_0(x)\) for which the exponents of the corresponding solutions do not exceed \(d_i+\varepsilon\), \(1\leq i\leq r+1\), is a regular image \(M(\{d_j\}_{j=1}^{i-1})\).

Corollary. Suppose there exist \(\varepsilon_n\to0\), \(T_n\to\infty\), \(\beta_n\to0\) such that

\[ \int_{T_n}^{t}\frac{e^{(\varepsilon_n/2)(\tau-t)}\gamma(\tau)}{(t-\tau)^\alpha}\,d\tau,\quad \int_t^\infty e^{(\varepsilon_n/2)(t-\tau)}\gamma(\tau)\,d\tau<\beta_n,\quad t>T_n,\quad n=1,2,\ldots . \]

In this case the exponent in \(\mathcal L_p(\Omega)\) (and hence \(u\) in \(C_k(\overline\Omega)\)) of any generalized solution of problem (1) coincides with one of \(d_i\), \(i=1,2,\ldots\).

The conditions of the corollary are satisfied if \(\lim_{t\to\infty}\gamma(t)=0\) or \(\gamma(t)\in \mathcal L_q(t_0,+\infty)\), \(q>\dfrac{1}{1-\alpha}\).

We shall consider equation (1) in \(\mathcal L_2(\Omega)\) and assume additionally that the operator \(\mathcal L\) and the boundary operators are formally self-adjoint. In this case, instead of requiring conditions 1, 2 to hold, it is sufficient to require that they hold for \(\theta=0\), and then \(A_2\) is the self-adjoint semibounded-from-above operator (1).

The following refinement of Theorem 2 is valid:

Theorem 3. Let an integer \(r\geq1\), \(\varepsilon>0\), \(\varepsilon<\min_{1\leq i\leq r}\dfrac{d_i-d_{i+1}}{2}\), \(2(2m-k)>n\), \(\alpha\in\left(\dfrac{k+n/2}{2m},1\right)\), be given, and suppose that

\[ \varphi_i(t)= \begin{cases} \displaystyle \alpha^\alpha\int_{t_0}^{t}\frac{e^{\varepsilon(\tau-t)}\gamma(\tau)\,d\tau}{(t-\tau)^\alpha}, & \displaystyle t\in\left[t_0,t_0+\frac{\alpha}{\lambda_0-d_i}\right], \\[2.2ex] \displaystyle \alpha^\alpha\int_{t-\frac{\alpha}{\lambda_0-d_i}}^{t} \frac{e^{\varepsilon(\tau-t)}\gamma(\tau)\,d\tau}{(t-\tau)^\alpha} +(\lambda_0-d_i)^\alpha \int_{t_0}^{\,t-\frac{\alpha}{\lambda_0-d_i}} e^{\varepsilon(\tau-t)}\gamma(\tau)\,d\tau, & \displaystyle t>t_0+\frac{\alpha}{\lambda_0-d_i}, \end{cases} \]

\[ \psi_i(t)=(\lambda_0-d_{i-1})^\alpha\int_t^\infty e^{\varepsilon(t-\tau)}\gamma(\tau)\,d\tau, \]

\[ R_i=\sup_{t>t_0}\varphi_i(t),\qquad Q_i=\sup_{t>t_0}\psi_i(t),\qquad i=1,\ldots,r+1,\quad Q_0=0, \]

then

\[ R_i+Q_i<1,\quad i=1,\ldots,r+1. \]

In this case the assertions of Theorem 2 are valid for \(p=2\).

The results obtained can be applied to the study of the stability of the zero solution of (1).

I express my gratitude to V. V. Nemytskii for his attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
4 VII 1964

References

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4. V. P. Il’in, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 53, 359 (1959).
5. K. V. Valikov, DAN, 158, No. 5 (1964).