Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

Abstract

In this paper, we study the global existence and the asymptotic behavior of the solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects (E) $$ \left\{ \begin{aligned} & \psi _t = - (1 - \alpha )\psi - \theta _x + \alpha \psi _{xx} , \\ & \theta _t = - (1 - \alpha )\theta + \nu \psi _x + 2\psi \theta _x + \alpha \theta _{xx} , \\ \end{aligned} \right. $$ with initial data (I) $$ (\psi ,\theta )(x,0) = (\psi _0 (x),\theta _0 (x)) \to (\psi _ \pm ,\theta _ \pm )\quad {\text{as}}\quad x \to \pm \infty , $$ where α and ν are positive constants such that α < 1,ν < α (1−α). Through constructing a correct function % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK % aacaGGOaGaamiEaiaacYcacaWG0bGaaiykaaaa!3BB3! $$\hat \theta (x,t)$$ defined by (2.13) and using the energy method, we show % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGZbGaaiyDaiaacchaaSqaaiaadIhacqGHiiIZtuuDJXwAK1uy0HMm % aeHbfv3ySLgzG0uy0HgiuD3BaGqbbiab-1risbqabaGccaGGOaWaaq % qaaeaacaGGOaGaeqiYdKNaaiilaiabeI7aXjaacMcacaGGOaGaamiE % aiaacYcacaWG0bGaaiykamaaeeaabaGaey4kaScacaGLhWoadaabba % qaaiaacIcacqaHipqEdaWgaaWcbaGaamiEaaqabaaakiaawEa7aiaa % cYcacqaH4oqCdaWgaaWcbaGaamiEaaqabaGccaGGPaGaaiikaiaadI % hacaGGSaGaamiDaiaacMcadaabbaqaaiaacMcacqGHsgIRcaaIWaaa % caGLhWoaaiaawEa7aaaa!6701! $$\mathop {\sup }\limits_{x \in \mathbb{R}} (\left| {(\psi ,\theta )(x,t)\left| + \right.\left| {(\psi _x } \right.,\theta _x )(x,t)\left| {) \to 0} \right.} \right.$$ as % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk % ziUkabg6HiLcaa!3A45! $$t \to \infty $$ and the solutions decay with exponential rates. The same problem is studied by Tang and Zhao [10] for the case of (ψ±, θ±) = (0,0).