Conservative quasilinear first-order laws with an infinite domain of dependence on the initial data. (English. Russian original) Zbl 0789.35039

Sov. Math., Dokl. 42, No. 2, 316-321 (1991); translation from Dokl. Akad. Nauk SSSR 314, No. 1, 79-84 (1990).
The authors investigate the Cauchy problem in the layer \(\Pi_ T= (0,T]\times \mathbb{R}^ n\) for the equation \[ u_ t+\text{div}_ x \varphi(u)=0, \quad \varphi(u)= (\varphi_ 1(u),\dots, \varphi_ n(u)), \quad u=u(t,x)\tag{1} \] with the initial condition \[ u(0,x)= u_ 0(x)\in L_ \infty (\mathbb{R}^ n), \quad x\in\mathbb{R}^ n, \quad | u_ 0(x)|\leq M_ 0= \text{ess sup }| u_ 0(x)|\tag{2} \] in a nonlocal formulation (here \(T\) is any positive number) and under the assumption only that the functions \(\varphi_ i(u)\) are continuous. The last assumption has the effect that the domain of dependence on the initial data is infinite. Existence and uniqueness of generalized solutions of (1)–(2) is stated. Schemes of proofs are given.


35F25 Initial value problems for nonlinear first-order PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs