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Counterexamples to Strichartz estimates for the wave equation in domains. (English) Zbl 1201.35060
For the solutions to the wave equation in the flat space $$\mathbb{R}^n$$
$(\partial_t^2-\Delta)u(t,x)=0, \quad u(0,x)=f, \quad \partial_t u(0,x)=g,\quad (t,x)\in \mathbb R\times\mathbb R^n,$
the Strichartz estimates are a family of space time integrability estimates and state that
$\|u\|_{L^q_t L^r_x}\leq C \big(\|f\|_{\dot H^s}+\|g\|_{\dot H^{s-1}}\big)$
for
$\frac{1}{q}\leq \frac{n-1}{2} \bigg(\frac{1}{2}-\frac{1}{r}\bigg), \quad q\geq 2, \quad (q,r)\neq \Bigg(\max \bigg\{2,\frac{4}{n-1}\bigg\},\infty\Bigg)$
and $$s=\frac{n}{2}-\frac{n}{r}-\frac{1}{q}$$ [see D. Fang and C. Wang, Nonlinear Anal., Theory Methods Appl. 65, No. 3 (A), 697–706 (2006; Zbl 1096.35026)].
There have been many works in proving the Strichartz estimates in various settings. If the domain is the exterior in $$\mathbb{R}^n$$ of a compact set with smooth nontrapping boundary, the estimates were established for the full range of exponents $$(q,r,s)$$ with $$q>2$$ and $$r<\infty$$ [see K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge and Y. Zhou, Trans. Am. Math. Soc. 362, No. 5, 2789–2809 (2010; Zbl 1193.35100)]. If the domain is a compact manifold without boundary, the local in time estimates have also been established for the full range of exponents $$(q,r,s)$$ with $$r<\infty$$ [see D. Tataru, J. Am. Math. Soc. 15, No. 2, 419–442 (2002; Zbl 0990.35027)]. For a manifold with boundary, together with either Dirichlet or Neumann homogeneous boundary conditions, there have been some works showing that the local in time Strichartz estimates hold for certain subset of the range of indices [see M. D. Blair, H. F. Smith and C. D. Sogge, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 5, 1817–1829 (2009; Zbl 1198.58012)].
In this work, for a sample 2-dimensional strictly convex domain with Dirichlet boundary condition, the author provide a counterexample to the local Strichartz estimates for
$\frac{3}{q}+\frac{1}{r}>\frac{15}{24}.$
More precisely, for $$(q,r)$$ with $$\frac{2}{q}=\frac{1}{2}-\frac{1}{r}$$ and $$4<r<\infty$$, the author proves that to have the local Strichartz estimates hold, we need to add at least $$\frac{1}{6}(\frac{1}{4}-\frac{1}{r})$$ on $$s$$.

##### MSC:
 35B45 A priori estimates in context of PDEs 35L05 Wave equation 35L20 Initial-boundary value problems for second-order hyperbolic equations
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##### References:
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