Logarithmic Klein-Gordon Equation

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Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term (2019, unknown): “Haraux and Cazenave[12] proved the existence and uniqueness of solution for the Cauchy problem for the nonlinear Schrödinger equation iut + ∆u + Vu + ku log |u|2 = 0 in RN ×R+ (1.11) and for the nonlinear Klein-Gordon equation utt − ∆u = ku log |u|2 in R3 ×R+. (1.12) Górka [13] obtained that the ini”
GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR A LOGARITHMIC WAVE EQUATION ARISING FROM Q-BALL DYNAMICS (2013, methodology): “In [12], by using compactness method Gorka obtained the global existence of weak solutions to the problem (1.”
Energy decay estimates and infinite blow‐up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source (2018, methodology): “(Weak solution) u(x, t) is called a weak solution of (1) to (3) on Ω × (0,T) if u ∈ C(0,T;H1 0(Ω)) with ut ∈ C(0,T;H 1 0(Ω)) and satisfies (1) to (3) in the distribution sense, ie, (utt, φ)L2(Ω) + (∇u,∇φ)L2(Ω) + (∇ut,∇φ)L2(Ω) = (u log |u|2, φ)L2(Ω), for any φ ∈ H1 0(Ω), t ∈ (0,T), where u(x, 0) = u0”
Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay (2018, methodology): “[5] Gorka P. Logarithmic Klein-Gordon equation.”