Some results on the well-posedness for second order linear equations (Q1043686)

The author considers the Cauchy problems for second order hyperbolic equations of the following form: \[ Lu=f(t,x), \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),\tag{1} \] where \(L\) is a differential, second order operator \[ Lu=u_{tt}- \sum_{i,j=1}^n(a_{ij}(t,x)u_{x_j})_{x_i}+\sum_{i=1}^n [(b_iu_{x_i})_t+ (b_iu_t)_{x_i}]+cu_t+ \sum_{i=1}^nd_iu_{x_i}+eu,\tag{2} \] where: \(a_{ij}=a_{ji}\), and all the coefficients of \(L\) are assumed to be real-valued \(C^\infty\) functions in the strip \( G_T:=[0,T]\times{\mathbb R}^n\), i.e. which belong to the class \(B^\infty(G_T,{\mathbb R})\). The author investigates the well-posedness the Cauchy problem (1) in the following sense: We say that (1) is well-posed (in \(C^\infty\)) when, for each \(\bar x\in{\mathbb R}^n\), there is some neighbourhood \(V\) of \((0,\bar x)\) in \(G_T\) such that there is a unique solution \(u\in C^\infty(V)\) for all \(u_0,u_1\in C^\infty({\mathbb R}^n)\) and all \(f\in C^\infty(G_T)\). If moreover, we can take \(V=G_T\), then we say that (1) is globally well-posed. The main result is the following theorem: Theorem. (i) For any hyperbolic operator of type (2) satisfying the condition \(b_i=0\) or \(\partial_{x_i}b_j=0\), the Cauchy problem is globally well-posed if, for some constants \(C,A>0\) one has \[ t\left[\sum_{i=1}^n(d_i-cb_i)\xi_i\right]^2\leq c\varphi_A(t,x,\xi), \tag{3} \] where: \[ \varphi_A=\sum_{i,j=1}^n\{A\Delta_{ij}+\partial_t\Delta_{ij}+\sum_{k=1}^n (b_n\partial_{x_n}\Delta_{ij}-2\Delta_{ij}\partial_{x_h}\partial_j\} \xi_i\xi_j, \tag{4} \] and \(\Delta_{ij}=b_ib_i+a_{ij}\) \((i,j=1,2,\dots,n)\). (ii) In absence of condition (4) i.e. for a hyperbolic operator of the general type (2) in order to get the well-posedness we must replace (3) with the stronger condition \[ t\left[\left[\sum_{i=1}^n(d_i-cb_i)\xi_i\right]^2+\rho(x,t)\right]\leq c\varphi_A(t,x), \qquad \rho=\sum_{i=1}^nA_{ij}(t,x)\xi_i^2. \] Moreover the well-posedness is no longer global, in general. Finally, the author presents a sufficient condition for the well posedness of \(2\times2\) systems in space dimension \(n=1\) with \(L=I\partial_t+A(t,x)\partial_x+B(t,x)\) under the assumtion \(A,B\in B^\infty(G_T,M_2({\mathbb R})\).