Global small data solutions for semilinear waves with two dissipative terms

Abstract

<jats:p>In this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity <jats:inline-formula><jats:alternatives><jats:tex-math>$$|u|^p$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msup>\n <mml:mrow>\n <mml:mo>|</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo>|</mml:mo>\n </mml:mrow>\n <mml:mi>p</mml:mi>\n </mml:msup>\n </mml:math></jats:alternatives></jats:inline-formula> or nonlinearity of derivative type <jats:inline-formula><jats:alternatives><jats:tex-math>$$|u_t|^p$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mrow>\n <mml:mo>|</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>u</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:msup>\n <mml:mrow>\n <mml:mo>|</mml:mo>\n </mml:mrow>\n <mml:mi>p</mml:mi>\n </mml:msup>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>, in any space dimension\xa0<jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\geqslant 1$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>⩾</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>, for supercritical powers\xa0<jats:inline-formula><jats:alternatives><jats:tex-math>$$p>{\\bar{p}}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>p</mml:mi>\n <mml:mo>></mml:mo>\n <mml:mover>\n <mml:mrow>\n <mml:mi>p</mml:mi>\n </mml:mrow>\n <mml:mrow>\n <mml:mo>¯</mml:mo>\n </mml:mrow>\n </mml:mover>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>. The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^r-L^q$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:msup>\n <mml:mi>L</mml:mi>\n <mml:mi>r</mml:mi>\n </mml:msup>\n <mml:mo>-</mml:mo>\n <mml:msup>\n <mml:mi>L</mml:mi>\n <mml:mi>q</mml:mi>\n </mml:msup>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> long time decay estimates for the solution in the full range <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\\leqslant r\\leqslant q\\leqslant \\infty $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>1</mml:mn>\n <mml:mo>⩽</mml:mo>\n <mml:mi>r</mml:mi>\n <mml:mo>⩽</mml:mo>\n <mml:mi>q</mml:mi>\n <mml:mo>⩽</mml:mo>\n <mml:mi>∞</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>. The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers\xa0<jats:inline-formula><jats:alternatives><jats:tex-math>$$p<{\\bar{p}}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>p</mml:mi>\n <mml:mo><</mml:mo>\n <mml:mover>\n <mml:mrow>\n <mml:mi>p</mml:mi>\n </mml:mrow>\n <mml:mrow>\n <mml:mo>¯</mml:mo>\n </mml:mrow>\n </mml:mover>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>