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Propagation of analyticity for a class of nonlinear hyperbolic equations. (English) Zbl 1225.35151

The author considers the Cauchy problem for a semilinear hyperbolic equation in \((t,x)\in [0,T]\times\mathbb R\) of the following form:
\[ Lu(t,x)=\partial_t^m u + a_1 (t)\partial_t^{m-1}\partial_x u + \dots + a_m (t)\partial_x^m u =f(u). \]
Here \(f(u)\) is a real entire function. One assumes that \(L\) is hyperbolic, i.e., that the characteristic roots \(\lambda_1 (t), \dots ,\lambda_m (t)\) are real functions. One admits weak hyperbolicity, but in such a case one assumes that these characteristic roots satisfy some uniformity condition.
For this Cauchy problem, the author gives two results about the regularity of the solution. First, he assumes that the coefficients \(a_j (t)\) are analytic (with respect to \(t\)). In this case, if the initial values \(u(0,\cdot ),\partial_t u(0,\cdot ), \dots ,\partial_t^{m-1} u(0,\cdot )\) are analytic (with respect to \(x\)), and \(u(t,\cdot ),\partial_t u(t,\cdot ), \dots\), \(\partial_t^{m-1} u(t,\cdot )\) have some “regularity” and some “boundedness”, then the solution \(u\) to the Cauchy problem is analytic with respect to \(x\). Secondly, he assumes that the coefficients \(a_j (t)\) are \(C^{\infty}\). Then one must strengthen the above “regularity” and ”boundedness” using Gevrey classes, but basically a similar result is true. These are natural generalizations of similar results for linear equations.

MSC:

35L76 Higher-order semilinear hyperbolic equations
35L30 Initial value problems for higher-order hyperbolic equations
35L80 Degenerate hyperbolic equations
35A21 Singularity in context of PDEs
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References:

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