Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for a nonhomogeneous quasilinear hyperbolic system
The authors of this interesting paper consider a nonhomogeneous quasilinear hyperbolic system \[ {\partial u\over \partial t}+A(u){\partial u\over \partial x}+Lu=0, \] where \(u=(u_1,u_2, \ldots , u_n)^T\) is the unknown vector function of \((t,x)\), \(A(u)\) is \(n\times n\) matrix with suitably smooth elements \(a_{ij}(u)\) (\(i,j=1,2,\ldots , n\)), \(L>0\) is a constant. The following decay initial data at \(t=0\) is given: \[ u=\varphi (x) \quad (x\geq 0), \] where \(\varphi (x)\) is a small \(C^1\) function of \(x\) with certain decay properties as \(x\to\infty \). The main result here is that there exists \(\theta_0>0\) so small that for any given \(\theta \in [0,\theta_0]\), the above stated Cauchy problem admits a unique global \(C^1\) solution \(u=u(t,x)\) on the domain \(D=\{(t,x): t\geq 0\), \(x\geq x_n(t)\}\), where \(x=x_n(t)\) is the \(n\)-th forward characteristic emanating from the origin \(O(0,0)\), \[ {d x_n(t)\over dt}=\lambda_n(u(t,x_n(t))), \qquad x_n(0)=0. \] (\(\lambda_n(u)\) is the \(n\)-th eigenvalue to the system.) Using the above stated theory the authors prove the existence of global classical, \(C^1\) solutions of the flow equations of a model class of fluids with viscosity induced by fading memory with small smooth initial data.
scientific article; zbMATH DE number 5078920
scientific article