Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach

In this paper, we study the following water wave model with a nonlocal viscous term: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>x</m:mi> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mfrac> <m:msqrt> <m:mi>ν</m:mi> </m:msqrt> <m:msqrt> <m:mi>π</m:mi> </m:msqrt> </m:mfrac> <m:mo>⁢</m:mo> <m:mfrac> <m:mo>∂</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mn>0</m:mn> <m:mi>t</m:mi> </m:msubsup> <m:mrow> <m:mpadded width="+1.7pt"> <m:mfrac> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:msqrt> <m:mrow> <m:mi>t</m:mi> <m:mo>-</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msqrt> </m:mfrac> </m:mpadded> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>s</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>x</m:mi> </m:msub> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>ν</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> uₜ+uₓ+β uₓₓₓ+{√ν}{√π}∂/% ∂ t∫₀ᵗ{u(s)}{√t-s}\,ds+uuₓ=ν uₓₓ, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:msqrt> <m:mi>π</m:mi> </m:msqrt> </m:mfrac> <m:mo>⁢</m:mo> <m:mfrac> <m:mo>∂</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mn>0</m:mn> <m:mi>t</m:mi> </m:msubsup> <m:mrow> <m:mpadded width="+1.7pt"> <m:mfrac> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:msqrt> <m:mrow> <m:mi>t</m:mi> <m:mo>-</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msqrt> </m:mfrac> </m:mpadded> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>s</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {{1}{√π}∂/∂ t∫₀ᵗ{u(s)}{√% t-s}\,ds} is the Riemann–Liouville half-order derivative. We prove the well-posedness of this model using diffusive realization of the half-order derivative, and we discuss the asymptotic convergence of the solution. Also, we compare our mathematical results with those given in [5] and [14] for similar equations.