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Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type. (English. Russian original) Zbl 1221.35005

J. Math. Sci., New York 148, No. 1, 1-142 (2008); translation from Sovrem. Mat. Prilozh. 40 (2006).
There is studied the Cauchy problem for the following equation with operator-valued coefficients:
\[ \frac{d}{dt}\left(A_0 u+ \sum^N_{j=1}A_j(u)\right) +L_i(u)=F(u),\qquad u(0)=u_0,\quad i=0,1. \]
Here, the operators \(A_0:V_0\to V_0^*\), \(A_j:V_j\to V_j^*\), \(F:W_0\to W_0^*\), \(L_0:W_1\to W^*_1\), \(L_1(u):=D(P(u))\); \(P(u):W_2\to W_3\), \(D:W_3\to W_4\subset V_0^*\),
\[ \begin{aligned} V:=\bigcap^N_{j=0} &V_j\subseteq V_k \subseteq \mathbb H \subseteq V^*_k \subseteq V^*,\qquad k=0,\dots , N,\\ &V\subseteq W_j\subseteq \mathbb H \subseteq W^*_j \subseteq V^*,\qquad j=0,1;\qquad V_0 \subseteq W_2, \end{aligned} \]
are bounded in the corresponding Banach spaces and satisfy a long list of conditions. The main conditions are the following “positivity” (or “nonnegativity”) conditions:
\[ \begin{alignedat}{2} \langle A_0u-A_0v,u-v\rangle_0&\geq m_0\|u-v\|^2_{V_0} &&\qquad \forall\,u,v\in V_0, \quad m_0>0;\\ \langle A_j(u), u\rangle_j&\geq m_j\|u\|_{V_j}^{p_j}, &&\qquad m_j=\text{const}>0, \quad p_j>2,\;j=1,\dots,N;\\ (F(u),u)_0&\geq m_F\|u\|^{q+2}_{W_0}, &&\qquad m_F=\text{const}>0,\quad q>0;\\ (L_0u-L_0v, u-v)_1&\geq d_1|u-v|^2_{W_1}, &&\qquad d_1=\text{const}>0;\\ \langle D(P(u)),u\rangle_0&=0 &&\qquad \forall\,u\in V_0. \end{alignedat} \]
By the Faedo-Galerkin method for arbitrary \(u_0\in V\) it is proved the existence of \(T_0=T_{u_0}>0\) such that the mentioned problem has a unique weak (strong) generalized solution \(u(t)\) on the interval \([0,T)\) for all \(T<T_0\). Additionally, if \(\alpha:=(q+2)\bar p^{-1}<1\), where \(\bar p:=\max_{j\leq N} p_j\), then \(T_0=\infty\). If \(\alpha>1\), then using some generalization of the H. Levine energy estimate method, it is proved that \(T_0=T_{u_0}<\infty\) for arbitrary “sufficiently large” \(u_0\). Estimates of \(T_0\) and the corresponding energy norms of \(u(t)\) are obtained, too.
Initial-boundary problems for many concrete different pseudoparabolic equations, covered by the general approach, are mentioned, in particular
\[ A_0=-\Delta,\quad A_j(u)=-\Delta_{p_j}(u),\quad L_0u =u,\quad F(u)=|u|^{q+2}u;\tag{1} \]
\[ \begin{aligned} &A_0=\Delta^2-\Delta,\quad A_1(u)=-\Delta_p(u),\\ &L_1(u)=D(P(u)),\;P(u)=u^2,\;D=\frac{\partial}{\partial x_1},\quad F(u)=-\Delta_4(u); \end{aligned}\tag{2} \]
where \(\Delta_p(u):=\text{div}(|\nabla u|^{p-2}\nabla u)\).

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B44 Blow-up in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
35K90 Abstract parabolic equations
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
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References:

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