On the controllability for the Khokhlov–Zabolotskaya–Kuznetsov-like equation

References

• Aanonsen , SI , Barkve , T , Tjotta , JN and Tjotta , S . 1984 . Distortion and harmonic generation in the nearfield of a finite amplitude sound beam . J. Acoust. Soc. Amer. , 75 ( 3 ) : 749 – 768 .
   Google Scholar
• Averkiou , MA , Lee , Y-S and Hamilton , MF . 1993 . Self-demodulation of amplitude- and frequency-modulated pulses in a thermoviscous fluid . J. Acoust. Soc. Amer. , 94 ( 5 ) : 2876 – 2883 .
   Google Scholar
• Averkiou , MA and Hamilton , MF . 1997 . Nonlinear distortion of short pulses radiated by plane and focused circular pistons . J. Acoust. Soc. Amer. , 102 ( 5 ) : 2539 – 2548 . Part 1
   PubMedGoogle Scholar
• Bakhvalov , NS , Zhileikin , YaM and Zabolotskaya , EA . 1987 . Nonlinear Theory of Sound Beams , New York : American Institute of Physics . (Nelineinaya Teoriya Zvukovih Puchkov, Nauka, Moscow, 1982)
   Google Scholar
• Lee , Y-S and Hamilton , MF . 1995 . Time-domain modeling of pulsed finite-amplitude sound beams . J. Acoust. Soc. Amer. , 97 ( 2 ) : 906 – 917 .
   Google Scholar
• Sanchez , D . 2005 . Long waves in ferromagnetic media, Zabolotskaya–Khokhlov equation . J. Differential Equ. , 210 : 263 – 289 .
   Google Scholar
• Zabolotskaya , EA and Khokhlov , RV . 1969 . Quasi-planes waves in the nonlinear acoustic of confined beams . Sov. Phys. Acoust. , 15 : 35 – 40 .
   Google Scholar
• Rozanova , AV . 2007 . Khokhlov–Zabolotskaya–Kuznetsov equation . C.R. Acad. Sci. Paris, Ser. I , 344 : 337 – 342 .
   Google Scholar
• Rozanova-Pierrat , A . 2008 . Qualitative Analysis of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation . Math. Models Methods Appl. Sci. , 18 ( 5 ) : 781 – 812 .
   Google Scholar
• Rozanova-Pierrat , A . 2009 . On the derivation of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media . Commun. Math. Sci. , 7 ( 3 ) : 679 – 718 .
   Google Scholar
• Kostin , I and Panasenko , G . Analysis and Homogenization of the Khokhlov–Zabolotskaya–Kuznetsov Type Equation, oral communication in the International Conference on Advanced Problems in Mechanics, June 2005, St Petersburg
   Google Scholar
• Kostin , I and Panasenko , G . 2006 . Khokhlov–Zabolotskaya–Kuznetsov type equation: Nonlinear acoustics in heterogeneous media . C.R. Mécanique , 334 ( 4 ) : 220 – 224 .
   Google Scholar
• Howard , SM and Zanelli , CI . 2007 . HIFU transducer characterization using a robust needle hydrophone, Proceedings of the International Symposium on Therapeutic Ultrasound, 30 August–2 September 2006 , Edited by: Coussios , C-C and ter Haar , G . Oxford, UK : American Institute of Physics .
   Google Scholar
• Liu , X , Lia , J , Gonga , X and Zhanga , D . 2006 . Nonlinear absorption in biological tissue for high intensity focused ultrasound . Ultrasonics , 44 ( 1 ) : e27 – e30 .
   PubMedGoogle Scholar
• Soneson , Joshua E and Myers , Matthew R . 2007 . Gaussian representation of high-intensity focused ultrasound beams . J. Acoust. Soc. Amer. , 122 ( 5 ) : 2526 – 2531 .
   PubMedGoogle Scholar
• Rozanova , AV . 2004 . Controllability in a nonlinear parabolic problem with integral overdetermination . Differential Equations , 40 ( 6 ) : 853 – 872 .
   Google Scholar
• Rozanova , AV . 2004 . Controllability for a nonlinear abstract evolution equation . Math. Notes , 76 ( 4 ) : 511 – 524 .
   Google Scholar
• Rozanova , AV . 2005 . Letter to the editor . Math. Notes , 78 ( 5–6 ) : 745
   Google Scholar
• Ladyzhenskaya , OA . 1985 . Kraevye Zadachi Matematicheskoi Fiziki (Boundary Value Problems of Mathematical Physics , New York, Berlin, London : Springer-Verlag . Nauka, Moscow, 1973
   Google Scholar
• Pazy , A . 1983 . Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences , Vol. 44 , Berlin, Heidelberg, New York : Springer .
   Google Scholar
• Kostin , AB . 1991 . “ Inverse Problem for the Heat equation with an Integral over-determination ” . In Obratnye Zadachi Dlya Matematicheskikh Modelei Fizicheskikh Protsessov (Inverse Problems for Mathematical Models of Physical Processes) , 45 – 48 . Moscow : Moscow Engineering Physics Institute .
   Google Scholar