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Weighted Strichartz estimates and existence of self-similar solutions for semilinear wave equations. (English) Zbl 1053.35029
This paper is devoted to the Cauchy problem for nonlinear wave equations with homogeneous nonlinearities of degree \(p\) in the unknown function, so that the equation is invariant under a suitably normalized dilation. The initial data are taken to be small, radial and homogeneous of the appropriate degree to be invariant under that dilation (they are therefore suitable powers of \(| x|\)) so as to generate self-similar solutions. The space dimension is supposed to be odd and not smaller than 3. The range of \(p\) is \(p_0(n)< p< p_c(n)= (n+ 3)/(n- 1)\), where \(p_0(n)\) is the known critical exponent for the existence of global solutions for small regular data, and \(p_c(n)\) is the conformal invariance value.
The main result is that under the previous assumptions, the Cauchy problem for the given equation has a unique solution in a suitably weighted weak \(L^{p+1}\) space. The weight has the form \(| t^2- x^2|^\gamma\), where \(\gamma\) is chosen so as to make the quasinorm invariant under the dilation adapted to the equation.
The proof uses in particular Strichartz estimates in such spaces which are of independent interest. These estimates are restricted to odd space dimension and to radial functions. They are proved through the use of a representation of the solution of the inhomogeneous equation in terms of Legendre polynomials and of real interpolation between \(L^q\) and weak \(L^q\) spaces.

MSC:
35B45 A priori estimates in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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