We studied a stochastic heat equation with proportional space-time delay and scalar multiplicative Brownian noise. The model described diffusion with scale-dependent memory, where delayed feedback was evaluated at spatially rescaled locations. A key feature was that the associated spatial pantograph operator is non-isometric, leading to an additional spatial-compression effect in the mean-square stability condition. We established well-posedness, mean-square boundedness, and algebraic mean-square decay under suitable stability assumptions. For the numerical approximation, we developed a semi-implicit finite-difference Euler-Maruyama method that treated diffusion and damping implicitly while evaluating the delayed and stochastic terms explicitly. Under appropriate interpolation and step-size assumptions, the scheme preserved mean-square boundedness and algebraic decay and achieved second-order spatial accuracy and the classical one-half strong temporal order. Numerical experiments supported the theoretical results, demonstrated the effectiveness of the semi-implicit treatment of stiffness, and quantified the influence of spatial compression on the stability margin.
Citation: Fathalla A. Rihan. Mean-square stability and semi-implicit Euler-Maruyama approximation of a stochastic pantograph heat equation with proportional space-time delay[J]. AIMS Mathematics, 2026, 11(9): 30233-30259. doi: 10.3934/math.20261198
We studied a stochastic heat equation with proportional space-time delay and scalar multiplicative Brownian noise. The model described diffusion with scale-dependent memory, where delayed feedback was evaluated at spatially rescaled locations. A key feature was that the associated spatial pantograph operator is non-isometric, leading to an additional spatial-compression effect in the mean-square stability condition. We established well-posedness, mean-square boundedness, and algebraic mean-square decay under suitable stability assumptions. For the numerical approximation, we developed a semi-implicit finite-difference Euler-Maruyama method that treated diffusion and damping implicitly while evaluating the delayed and stochastic terms explicitly. Under appropriate interpolation and step-size assumptions, the scheme preserved mean-square boundedness and algebraic decay and achieved second-order spatial accuracy and the classical one-half strong temporal order. Numerical experiments supported the theoretical results, demonstrated the effectiveness of the semi-implicit treatment of stiffness, and quantified the influence of spatial compression on the stability margin.
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Fathalla A. Rihan. Mean-square stability and semi-implicit Euler-Maruyama approximation of a stochastic pantograph heat equation with proportional space-time delay[J]. AIMS Mathematics, 2026, 11(9): 30233-30259. doi: 10.3934/math.20261198
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