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The Cauchy problem for quasilinear equations of odd order. (Russian) Zbl 0701.35046

The author considers the Cauchy problem of the equation \[ (1)\quad A(t,x)D_ tu-\sum_{0\leq | \alpha | \leq 2k+1}A_{\alpha}(t,x)D^{\alpha}_ xu=\sum_{0\leq \alpha \leq k}(- 1)^{\alpha}D^{\alpha}_ x[g_{\alpha}(t,x,\delta_ x^{\delta - 1}u)]+F(t,x), \] for \((t,x)\in \Pi_ T=(0,T)\times R^ n\) with the initial condition (2) \(u(0,x)=u_ 0(x)\) for \(x\in R^ n\). The author shows theorems for existence, uniqueness and continuous dependence on initial condition of generalized solution of the Cauchy problem (1) and (2) with nonregular initial condition \(u_ 0(x)\in L_ 2(R^ n)\).
Reviewer: J.Wang

MSC:

35D10 Regularity of generalized solutions of PDE (MSC2000)
35K55 Nonlinear parabolic equations
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