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Well-posedness in Sobolev spaces for semi-linear 3-evolution equations. (English) Zbl 1311.35061

Let \(D=-i\partial\). Consider the semilinear Cauchy problem \[ (D_t+A(t,x,u,D_x))u=f(t,x),\,\,\,\,(t,x)\in[0,T]\times\mathbb{R},\quad u(0,x)=u_0(x),\,\,x\in\mathbb{R}, \] where \(A\) is a third order differential operator of the form (\(w\in\mathbb{C}\)) \[ A(t,x,w,D_x)=a_3(t)D_x^3+a_2(t,x,w)D_x^2+a_1(t,x,w)D_x+a_0(t,x,w), \] where \(a_3\) is real, continuous and bounded from below by a positive constant, and the other coefficients \(a_2\) to \(a_0\) are complex valued, continuous in \(t\) and smooth in \((x,w)\), and such that as functions of \(x\) are bounded on \(\mathbb{R}\) along with the derivatives of all orders. Under suitable decay assumptions as \(|x|\to+\infty\) on Re\((a_2)\), Im\((a_2)\), \(\partial_w a_2\), \(\partial_x\)Re\((a_2)\) and Im\((a_1)\) the authors prove that the above Cauchy problem is locally in time \(H^\infty\) well-posed.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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